Subsubsection 3.1.2 Multiplicative theory
Definition 3.6 Unit group
\begin{equation*}
U_v = \ker(x\mapsto |x|_v)
\end{equation*}
it is compact, open if \(v\) is non-archimidean, e.g.
\begin{equation*}
U_v = \begin{cases}S^1 \amp \text{ if }K_v = \CC \\S^1 \amp \text{ if }K_v = \RR \\ \ints_v^\times \amp\text{ if } v\text{ non-archimidean}\end{cases}\text{.}
\end{equation*}
Definition 3.7
A quasicharacter is
\begin{equation*}
K_v^\times \to \CC^\times
\end{equation*}
A (unitary) character is
\begin{equation*}
K_v^\times \to S^1\text{.}
\end{equation*}
Such a map is unramified if it is trivial on \(U_v\text{.}\) E.g.
\begin{equation*}
\xi \mapsto |\xi|_v^s
\end{equation*}
is unramified \(s\in \CC\text{.}\)
Lemma 3.8
All unramified characters are of this form
\begin{equation*}
\widehat K_\nr^\times= \CC/ (2\pi i/ \log(N(\varpi_v)))
\end{equation*}
if \(v\) is non-archimidean.
Choose a uniformiser \(\varpi_v\) s.t.
\begin{equation*}
\begin{array}{c|c|c} \alpha \amp \tilde \alpha \amp \varpi_v^{v_v(\alpha)} \\ \hline \CC^\times \amp S^1 \amp \RR_+ \\ \RR^\times \amp \{\pm 1\}\amp \RR_+ \\ K_v^\times \amp \ints_v^\times \amp \ZZ\end{array}\text{.}
\end{equation*}
Theorem 3.9
All quasicharacters of \(K_v^\times\) are of the form
\begin{equation*}
\alpha \mapsto c(\alpha) = \tilde c (\tilde \alpha) |\alpha|^s
\end{equation*}
where
\begin{equation*}
\tilde c \colon U_v \to \CC^\times\text{.}
\end{equation*}
Proof
Example 3.10
Dirichlet characters mod \(p\text{.}\)
E.g. over \(\CC^\times\) then
\begin{equation*}
\tilde c(\alpha) = \left(\frac{\alpha}{|\alpha|}\right)^m,\,m\in \ZZ
\end{equation*}
over \(\RR^\times\) then
\begin{equation*}
\tilde c(\alpha) = \left(\frac{\alpha}{|\alpha|}\right)^m,\,m\in \{0,1\}
\end{equation*}
over \(\ints_v^\times\) then
\begin{equation*}
\tilde c(\alpha)|_{1+\varpi^k\ints_v} \equiv 1
\end{equation*}
because \(c\) is a continuous map from a \(p\)-adic field to the complex numbers. Let
\begin{equation*}
\tilde k = \min\{k\in \NN : \tilde c(\alpha)|_{1+ \varpi_v^k \ints_v} \equiv 1\}
\end{equation*}
and \(\varpi_v^{\tilde k} = f_v\) is the conductor of \(\tilde c\text{.}\)
Multiplicative Haar measures
Let
\begin{equation*}
\diff_v^\times \alpha = \begin{cases} \frac{\diff_v \alpha}{|\alpha|_v} \amp \text{ if } v \text{ archimidean}\\ \left(\frac{1}{1-1/N(\varpi_v)}\right) \frac{\diff_v \alpha}{|\alpha|_v} \amp \text{ if } v \text{ non-archimidean}\end{cases}
\end{equation*}
where \(\diff_v \alpha\) is the additive Haar measure. these extra factors are really Tamagawa numbers. They make the product in the next lemma converge.
Lemma 3.13
For \(v\) non-archimidean
\begin{equation*}
\int_{\ints_v^\times } \diff_v^\times \alpha = N(D_v)^{-1/2}\text{.}
\end{equation*}
Proof
\begin{equation*}
\int_{\ints_v^\times} \diff_v^\times \alpha= \int_{\ints_v^\times} \frac{\diff_v \alpha}{|\alpha|_v} (1- N(\varpi_v)\inv)\inv
\end{equation*}
\begin{equation*}
= \int_{\ints_v^\times} \diff_v \alpha (1- N(\varpi_v)\inv)\inv
\end{equation*}
\begin{equation*}
= \sum_{\beta \in (\ints_v/\varpi_v\ints_v)^\times} \int _{\beta + \varpi_v\ints_v} \diff_v \alpha(1- N(\varpi_v)\inv)\inv
\end{equation*}
\begin{equation*}
= |\varpi_v|_v \sum_{\beta \in (\ints_v/\varpi_v\ints_v)^\times} \int _{\ints_v} \diff_v \alpha(1- N(\varpi_v)\inv)\inv
\end{equation*}
\begin{equation*}
= N(D_v)^{-1/2} |\varpi_v| (N(\varpi_v) - 1) (1- N(\varpi_v)\inv)\inv = N(D_v)^{-1/2}\text{.}
\end{equation*}
We are trying to set up a general machinery that will take a quasicharacter and associate a zeta function. In fact we want have
\begin{equation*}
c\colon N_v^\times \to \CC^\times \leadsto \zeta(f,c)
\end{equation*}
a family of \(\zeta\)-functions. We will then look at the gcd over all possible \(f\text{,}\) this will be the \(L\)-factor.
Subsubsection 3.1.5 Explicit \(\zeta\) functions
First let \(K_v = \RR\) then we will use the following notation: Additively \(\xi\) with \(\Lambda(\xi) = -\xi\text{,}\) \(\diff \xi\) the Lebesgue measure. Multiplicatively \(\alpha\) with \(|\alpha|_v = |\alpha|_\RR\) and \(\diff \units \alpha = \diff \alpha/|\alpha|_\RR\text{.}\) We will use characters \(|\cdot |^s\) or \(\mathrm{sgn}|\cdot|^s\text{,}\) \(f_{|\cdot|^s}= e^{-\pi \xi^2}\) and \(f_{|\cdot|^s\mathrm{sgn}} (\xi) = \xi e^{-\pi \xi^2}\text{.}\) These have fourier transforms
\begin{equation*}
\hat f_{|\cdot|^s}(\xi) = f(\xi)
\end{equation*}
and
\begin{equation*}
\hat f_{|\cdot|^s\mathrm{sgn}}(\xi) = i f_\mathrm{sgn}(\xi)\text{.}
\end{equation*}
So that
\begin{equation*}
\zeta(f_{|\cdot|^s}, |\cdot|^s) = \int_{\RR\units} f(\alpha) |\alpha|^s \diff \units \alpha = \int_{\RR\units} e^{-\pi \alpha^2} | \alpha| ^s \diff\units \alpha
\end{equation*}
\begin{equation*}
= 2 \int_0^\infty e^{-\pi \alpha^2}
\end{equation*}
\begin{equation*}
= \pi^{-s/2} \Gamma\left(\frac s2\right)\text{.}
\end{equation*}
\begin{equation*}
\zeta(f_{|\cdot|^s\mathrm{sgn}}, |\cdot|^s\mathrm{sgn}) = \pi^{-(s+1)/2} \Gamma\left(\frac{s+1}{2}\right)
\end{equation*}
now
\begin{equation*}
\rho(|\cdot|^s) = 2^{1-s} \pi^{-s} \cos(\pi s /2) \Gamma(s)
\end{equation*}
\begin{equation*}
\rho(|\cdot|^s\mathrm{sgn}) = -i 2^{1-s} \pi^{-s} \sin(\pi s /2) \Gamma(s)\text{.}
\end{equation*}
Normalizing by \(s= 1/2\) we get \(1 \) and \(- i\) respectively.
Over \(\CC\)
\begin{equation*}
\begin{array}{cc}
K_v^+ \amp K_v\units\\
\hline
\xi = x+iy \amp \alpha = re^{i\theta}\\
\Lambda = -2x \amp |\alpha| = r^2\\
\diff \xi =2 |\diff x \diff y| \amp \diff\units \alpha =\diff \alpha /|\alpha|= 2|\diff r \diff \theta|/r\\
\end{array}
\end{equation*}
characters
\begin{equation*}
c \leadsto c_n,\,n\in\ZZ
\end{equation*}
\begin{equation*}
c_n(\alpha) = r^n e^{in \theta}
\end{equation*}
equivalence class
\begin{equation*}
\{c_n(\alpha) |\alpha|^s : s\in \CC \}\text{.}
\end{equation*}
Functions
\begin{equation*}
f_n(\xi) = \begin{cases} (x-iy)^n e^{-2\pi(x^2 + y^2)} \amp \text{ if } n \ge 0 \\ (x+iy)^{-n} e^{-2\pi(x^2 + y^2)} \amp \text{ if } n \le 0\end{cases}
\end{equation*}
\begin{equation*}
= \begin{cases} r^{|n|} e^{-in \theta} e^{-2\pi r^2} \amp\text{ if } n \ge 0 \\ r^{|n|} e^{in\theta} e^{-2\pi r^2}\amp \text{ if } n \le 0\end{cases}
\end{equation*}
fourier transforms
Claim 3.20
\begin{equation*}
\hat f_n(\xi) = i^{|n|} f_{-n}(\xi) \,\forall n
\end{equation*}
Proof
Induction on \(n\text{:}\)
\(n = 0\)
\begin{equation*}
f_0(\xi) = e^{-\pi (x^2 + y^2)}
\end{equation*}
\begin{equation*}
\hat f_0(\xi) = 2\int_{\CC} f_0(u+iv) \overbrace{e(2(ux-vy))}^{\Lambda((u+iv)(x+iy))} \diff u \diff v
\end{equation*}
\begin{equation*}
2\int_{\RR} e^{-2\pi (u^2 - 2ixu)} \diff u \int_\RR e^{-2\pi(v^2 + 2ivy)} \diff v
\end{equation*}
which by completing the square and Cauchy gives
\begin{equation*}
e^{-2\pi(x^2 + y^2)} = \hat f_0(\xi)
\end{equation*}
Assume \(\hat f_n(\xi) = i^n f_{-n}(\xi)\) for \(n \gt 0\text{.}\) Then the induction step is to use \(\diff / \diff \bar \xi\) in the integral defining \(\hat f_n(\xi)\) (exercise).
\begin{equation*}
\zeta(f_n, c_n |\cdot|^s) = \int_{\CC\units} f_n(\alpha) c_n(\alpha) |\alpha|^s\diff\units\alpha
\end{equation*}
\begin{equation*}
= 2 \int |r|^n e^{-i n \theta} e^{-2\pi r^2} e^{in \theta} r^{2s-2}
\end{equation*}
\begin{equation*}
= 4\pi \int_0^\infty r^{|n| + 2s - 2} e^{-2 \pi r^2} r \diff r
\end{equation*}
\begin{equation*}
= (2\pi)^{1 - s - |n|/2} \Gamma\left(s+ \frac{|n|}{2}\right)
\end{equation*}
now
\begin{equation*}
\zeta(f_{-n}, c_{-n}|\cdot|^s) = i^{|n|} (2\pi)^{s- |n|/2} \Gamma\left(1 - s + \frac{|n|}{2}\right)
\end{equation*}
\begin{equation*}
\rho(c_n|\cdot |^s) = \frac{(-i)^{|n|} (2 \pi)^{k-s}}{(2\pi)^s} \frac{\Gamma(s + \frac{|n|}{2})}{\Gamma(1-s + \frac{|n|}{2})}\text{.}
\end{equation*}
For \(K\) non-archimidean with \(|\varpi|_v= \frac 1q\text{.}\)
\begin{equation*}
\begin{array}{cc}
K^+ \amp K\units\\
\hline
\xi \amp \alpha = \tilde \alpha \varpi^{v(\alpha)}\\
\Lambda (\xi) = \lambda(\tr_{K/\QQ_p}(\xi)) \amp |\alpha| = q_v^{-v(\alpha)}\\
\diff \xi \implies \int_\ints \diff \xi = N(D_v)^{-1/2} \amp \diff\units \alpha =(1-q_v\inv)\diff \alpha /|\alpha| \implies \int_{\ints\units} \diff\units \alpha = N(D_v)^{-1/2}
\end{array}
\end{equation*}
For \(D_v\) the different ideal.
Quasicharacters
\begin{equation*}
c_n \colon \ints\units \to \CC\units
\end{equation*}
of conductor \(f (\varpi^n)\ints_v\text{.}\) \(c_n(\varpi) = 1\text{.}\) Equivalence class of \(c_n\)
\begin{equation*}
\{c_n |\cdot|^s : s\in \CC\}\text{.}
\end{equation*}
Functions
\begin{equation*}
f_n(\xi) = \begin{cases} e(\Lambda(\xi)) = e^{2\pi i \lambda(\tr(\xi))} \amp \text{ if } \xi\in D_v\inv \varpi^{-n} \\ 0 \amp\end{cases}\text{.}
\end{equation*}
The fourier transforms
Lemma 3.21
\begin{equation*}
\hat f_n(\xi) = \begin{cases} |D_v|^{-1/2} |\varpi_v|^{-n} \amp \text{ if } \xi \equiv 1 \pmod{\varpi^n} \\ 0 \amp \end{cases}
\end{equation*}
Proof
\begin{equation*}
\hat f_n(\xi) = \int_K f_n(\eta) e(-\Lambda (\eta \xi)) \diff \eta
\end{equation*}
\begin{equation*}
=\int_{D_v\inv \varpi^{-n} \ints_v} e(\Lambda(\eta(1-\xi))) \diff \eta
\end{equation*}
\begin{equation*}
= \begin{cases} 0 \amp \text{ if } \xi \ne 1 \pmod{\varpi^n} \\ |D_v|\inv |\varpi|^{-n} \int_{\ints_v} \diff \eta \amp \end{cases}
\end{equation*}
\begin{equation*}
= \begin{cases} 0 \amp \text{ if } \xi \ne 1 \pmod{\varpi^n} \\ |D_v|^{-1/2} |\varpi|^{-n} \amp \end{cases}
\end{equation*}
Unramified calculation:
\begin{equation*}
\zeta(f_0, |\cdot |^s) = \int_{K\units} f_0(\alpha) |\alpha|^s \diff\units \alpha
\end{equation*}
\begin{equation*}
= \int_{D_v\inv} e(\Lambda(\alpha) ) |\alpha| ^s \diff\units \alpha
\end{equation*}
\begin{equation*}
= \int_{D_v\inv} |\alpha|^s \diff\units \alpha
\end{equation*}
\begin{equation*}
= \sum_{k=0}^\infty|D_v|^{-s} \int_{\varpi^n \ints\units}|\alpha|^s \diff\units \alpha
\end{equation*}
\begin{equation*}
\left(\ints = \bigsqcup_{n=0}^\infty \varpi^n \ints\units_v\right)
\end{equation*}
\begin{equation*}
= \sum_{k=0}^\infty |D_v|^{-s+1/2} q_v^{-ns}
\end{equation*}
\begin{equation*}
= |D_v|^{-s+1/2} \frac{1}{1-1/q_v^s}
\end{equation*}
\begin{equation*}
\zeta(\hat f_0, |\cdot |^{1-s}) = \zeta(|D_v|^{-1/2} \mathbf 1_{\ints_v}, |\cdot |^{1-s})
\end{equation*}
\begin{equation*}
= N(D_v)^{1/2} \int_{\ints_v} |\alpha|^{1-s} \diff\units \alpha
\end{equation*}
\begin{equation*}
= \frac{1}{1-1/q_v^{1-s}}
\end{equation*}
Let's recap a little, we have \(K\units \hookrightarrow K^+\) and moreover \(K\units \acts K^+\) with two orbits \(\{0\}\) and the rest. Looking at function spaces we have
\begin{equation*}
\cinf_c(K\units) \hookrightarrow S(K^+)\text{,}
\end{equation*}
Schwartz functions on the right. Taking duals to get spaces of distributions
\begin{equation*}
1 \to ?? \to D(K^+)_c \to \cinf_c (K\units)^\vee_c \to 0\text{.}
\end{equation*}
Corollary 3.22
There is \(\rho(c)\) s.t.
\begin{equation*}
\zeta(f,c) = \rho(c) \zeta(\hat f, \hat c)
\end{equation*}
with
\begin{equation*}
\rho(c) = \frac{\zeta(f,c)}{\zeta(\hat f, \hat c)}\text{.}
\end{equation*}
We have done the case of unramified characters.
Case II: \(n \gt 0\)
\begin{equation*}
\zeta(f_n, c_n|\cdot|^s) = \int_{K_v\units} f_n(\alpha) c_n(\alpha) |\alpha|^s \diff\units \alpha
\end{equation*}
\begin{equation*}
= \int_{(D_vf_v) \inv} e(\Lambda(\alpha))(c(\alpha)) |\alpha|_v^s \diff \units \alpha\text{.}
\end{equation*}
Let
\begin{equation*}
(D_v) = (\varpi_v^d)
\end{equation*}
\begin{equation*}
f= (\varpi_v^n)
\end{equation*}
\begin{equation*}
= \int_{(\varpi_v^{n+d})\inv} e(\Lambda(\alpha)) c_n(\alpha) |\alpha|^s \diff \units \alpha
\end{equation*}
\begin{equation*}
= \sum_{j = -n-d}^\infty \int_{\varpi_v^j \ints_v\units } e(\Lambda(\alpha)) c_n(\alpha) |\alpha|^s \diff \units \alpha
\end{equation*}
let \(\alpha = \varpi_v^j \alpha\) which implies
\begin{equation*}
= \sum_{j = -n-d}^\infty q_v^{js}\int_{\ints_v\units } e(\Lambda(\varpi^j\alpha)) c_n(\alpha) |\alpha|^s \diff \units \alpha\text{.}
\end{equation*}
Lemma 3.23
If \(j \ge -d\) then
\begin{equation*}
\int_{\ints_v\units} (\Lambda(\varpi_v^j \alpha)) c_n(\alpha) \diff\units \alpha = 0\text{.}
\end{equation*}
Proof
\(j \ge -d\) implies
\begin{equation*}
\varpi_v^j \ints_v\units \subseteq D_v\inv
\end{equation*}
\begin{equation*}
\implies \Lambda(\varpi_v^j \ints_v\units) \subseteq\ZZ
\end{equation*}
\begin{equation*}
\implies e(\Lambda(\varpi_v^j \ints_v\units)) = 1
\end{equation*}
so the integral in question
\begin{equation*}
= \int_{\ints_v\units} c_n(\alpha) \diff\units \alpha
\end{equation*}
as \(c_n \ne 1\text{.}\)
So finitely many are non-zero.
Lemma 3.24
\(- n - d \lt j \lt -d\) then
\begin{equation*}
\int_{\ints_v\units} e(\Lambda(\varpi_v^j \alpha)) c_n(\alpha) \diff \units \alpha = 0\text{.}
\end{equation*}
Proof
(Note \(-n-d \lt j \lt -d \iff 0 \lt -j-d \lt n\)) Let \(\alpha = a (1 + \varpi_v^{-d-j} \alpha_1)\text{.}\) So we may write
\begin{equation*}
\sum_{a \mod{\units \varpi_v^{d+j}}} \int_{\ints_v} e(\Lambda(\varpi_v^ja (1+ \varpi_v^{-d-j} \alpha_1))) c_n(a (1+ \varpi_v^{-d-j} \alpha_1)) \diff \units \alpha\text{.}
\end{equation*}
(Note: \(\varpi_v^j (a(1+ \varpi_v^{-j-d} \alpha_1)) = a\varpi_v^j + a \varpi_v^{-d} \alpha_1\text{,}\) the last term is in \(D_v\inv\) so \(e(\Lambda(\varpi_v^ja (1+ \varpi_v^{-d-j} \alpha_1))) = e(\Lambda(\varpi_v^ja))\)) So the integral is
\begin{equation*}
\sum_{a \mod{\units \varpi_v^{d+j}}} c_n(a) e(\Lambda(\varpi_v^ja)) \int_{\ints_v} c_n(1+ \varpi_v^{-d-j} \alpha_1) \diff \units \alpha = 0
\end{equation*}
as we are integrating over a multiplicative group.
Therefore: In the ramified case we have
\begin{equation*}
\zeta(f_b, |\cdot |^s c_n) = e_v^{(n+d) s} \int_{\ints_v\units} e(\Lambda(\varpi_v^{-(n+d)} \alpha)) c_n(\alpha) \diff \units \alpha
\end{equation*}
\begin{equation*}
= q_v^{(n+d)s} \sum_{a \mod{\units \varpi_v^{ n }}} e(\Lambda(\varpi_v^{-(n+d)} a)) c_n(a) \int_{1 + \varpi_v^n \ints_v} \diff \units \alpha\text{.}
\end{equation*}
\begin{equation*}
N(D_v f_v)^s A_n G(c_n)
\end{equation*}
where
\begin{equation*}
G(c_n) = \sum_{a \mod{\units \varpi_v^{ n }}} e(\Lambda(a\varpi^{-d-n})) c_n(a)
\end{equation*}
is a Gauss-sum for \(c_n\text{,}\) with \(|G(c_n)| = N(f)^{1/2}\text{.}\)
Corollary 3.25
\begin{equation*}
\zeta(\hat f, c_n\inv |\cdot |^{1-s}) = N(D_v)^{1/2} N(f_v) A_n\text{.}
\end{equation*}
Proof
This finishes the proof of the local functional equation.
\begin{equation*}
\zeta(f,c) = \int_{K_v\units} f(\alpha) c(\alpha)\diff \units \alpha\text{.}
\end{equation*}
A little bit of a more modern approach that may be helpful if you want to read things since Tate.
\begin{equation*}
\psi_v \text{ additive character}
\end{equation*}
\begin{equation*}
\mu \text{ additive measure, self-dual w.r.t } \psi_v
\end{equation*}
Definition 3.26
A family of \(\zeta\)-integrals for each \(|\cdot |^s \chi_v\)
\begin{equation*}
\chi_v \colon K_v\units \to \CC \units\text{.}
\end{equation*}
\begin{equation*}
Z(s,\chi_v, f_v, \psi_v) = \int_{K_v\units} f(\alpha) \chi_v(\alpha) |\alpha|^s \diff \units \alpha
\end{equation*}
\begin{equation*}
Z \colon \mathcal S(K_v\units) \to \CC\text{.}
\end{equation*}
To study this around 0
\begin{equation*}
f(\alpha) = a\phi_1( \alpha) + \phi_2(\alpha)
\end{equation*}
where \(\phi_1\) is the characteristic function of a very small (depending on \(f\)) neighbourhood of 0. \(\phi_2\) is 0 around a sufficiently small neighbourhood of 0.
\begin{equation*}
Z(s, \chi_v, \phi_2, \psi_v) = \int_{K_v\units} \phi_2(\alpha) \chi_v(\alpha) |\alpha|_v^s \diff\units \alpha
\end{equation*}
\begin{equation*}
= \sum_{j = -N_1}^{N_2} c_j \int_{\varpi_v^j \ints_v\units} \chi_v(\alpha) |\alpha|^s \diff\units \alpha
\end{equation*}
This converges for all \(s \in \CC\) and lands in \(\CC\lb q_v^s, q_v^{-1}\rb\) .
\begin{equation*}
Z(s, \chi_v, \phi_1, \psi_v) = \sum_{j = M }^\infty \int_{\varpi_v^j \ints_v\units} \chi_v(\alpha) |\alpha|^s \diff\units \alpha
\end{equation*}
\begin{equation*}
= \sum_{j = M }^\infty | \varpi_v^j|^s \int_{\ints_v\units} \chi_v(\varpi_v^j\alpha) \diff\units \alpha
\end{equation*}
this is a multiple of
\begin{equation*}
\begin{cases} 0 \amp \text{ if } \chi_v \text{ is ram.} \\ \frac{1}{1 - 1 /q_v^s} \amp \text{ if } \chi_v \text{ unram.}\end{cases}\text{.}
\end{equation*}
Usually these observations are packed into saying
\begin{equation*}
Z(s,\chi_v, f)
\end{equation*}
is
- Rational function of \(q_v^s\) and in \(\CC\lb q_v^s, q_v^{-s}\rb\text{.}\)
- \(\chi_v\) is unramified if its entire.
- There exists \(f\in \mathcal S(K)\) s.t.
\begin{equation*}
Z(s,\chi, f) = L(s, \chi)\text{.}
\end{equation*}
- For all \(f\)
\begin{equation*}
\frac{Z(s, \chi, f)}{L(s, \chi, f)}
\end{equation*}
is entire.
It is then said that \(L(s,\chi)\) is the GCD of the \(\zeta\) integrals \(Z(s, \chi, f)\text{.}\)
\begin{equation*}
\begin{array}{c|c|c}
\text{Field} \amp \chi_v \amp L_v(s, \chi_v)\\
\hline
\CC \amp |\cdot|^s_\CC\chi_n \amp 2(2\pi)^{-(s+ |n|/2)} \Gamma(s + |n|/2)\\
\RR \amp |\cdot|^s_\RR \amp (\pi)^{-s/2)} \Gamma(s/2)\\
\RR \amp |\cdot|^s_\RR\mathrm{sgn} \amp \pi^{-(s+1)/2)} \Gamma((s+1)/2)\\
K_v \amp |\cdot|^s \chi_v \text{ ur} \amp (1- \chi_v( \varpi_v)/q_v^s)\inv\\
K_v \amp |\cdot|^s \chi_v \text{ ram}\amp 1\\
\end{array}
\end{equation*}