
\begin{proof}~
    
Pour $\Re(s) > 0$, on a :
$$
\begin{aligned}
\mathcal{Z}(g, \chi) &= \int_{F^*} g(x) \chi(x) \, dx^{\times} \\
&= \overset{\text{nuls si } k < n_\psi}{\sum_{k=-\infty}^{+\infty}} \int_{\varpi^k \OO^{\times}} \1_{\varpi^{n_\psi} \OO}(x) \psi(x) \overbrace{\chi(x)}^{=|x|^s=q^{-ks}} \, dx^{\times} \\
&= \sum_{k=n_\psi}^{+\infty} q^{-k s} \int_{\varpi^k \OO^{\times}} \, dx^{\times} \\
&= \frac{q^{-n_\psi s}}{1-q^{-s}} \operatorname{Vol}\left(\OO^{\times}, dx^{\times}\right) .
\end{aligned}
$$
\end{proof}

\begin{lemme}~
\begin{aligned}[t]
    \text{Si } n_\chi=0 \quad  \hat{g}&= | \OO / \mathcal{D} |^{-1/2} q^{-n_\psi} \1_{\OO}, \\
    \text{Si } n_\chi>0  \quad \hat{g}&= | \OO / \mathcal{D} |^{-1/2} q^{n_\chi-n_\psi} \1_{-U_{n_\chi}}.
\end{aligned}
\end{lemme}


\begin{proof}~
$\hat{g}(y)=\int_F g(x) \psi(x y) d x=\int_{\varpi^{n_\psi-n_\chi} \OO} \psi(x) \psi(x y) d x$
$$
=\int_{\varpi^{n_\psi-n_\chi} \OO} \psi(x(y+1)) d x
$$
Or $\operatorname{Cond}(\psi)=\varpi^{n_\psi} \OO$ et $\operatorname{Cond}\left(\psi_{y+1}\right)=(y+1)^{-1} \varpi^{n_\psi} \OO$. \\
L'intégrale est nulle sauf si $\varpi^{n_\psi-n_\chi} \OO \subseteq (y+1)^{-1} \varpi^{n_\psi} \OO$, c'est-à-dire, sauf si $y+1 \in \varpi^{n_\chi} \OO$, c'est-à-dire sauf si $y \in -U_{n_\chi}$. \\

Ainsi \begin{aligned}[t]
\hat{g}(y) & =\1_{-U_{n_\chi}}(y) \cdot \operatorname{Vol}\left(\varpi^{n_\psi-n_\chi} \OO, dx\right) \\
& =\1_{-U_{n_\chi}}(y) \cdot|\varpi|^{n_\psi-n_\chi} \quad \operatorname{Vol}(\OO, dx) \\
& =\1_{-U_{n_\chi}}(y) \cdot q^{n_\chi-n_\psi} \cdot| \OO / \mathcal{D} |^{-1/2} 
\end{aligned}
\ \\
\qedhere
\end{proof}



\begin{proof}~
    
Pour $\Re(s) > 0$, on a :
$$
\begin{aligned}
\mathcal{Z}(g, \chi) &= \int_{F^*} g(x) \chi(x) \, dx^{\times} \\
&= \overset{\text{nuls si } k < n_\psi}{\sum_{k=-\infty}^{+\infty}} \int_{\varpi^k \OO^{\times}} \1_{\varpi^{n_\psi} \OO}(x) \psi(x) \overbrace{\chi(x)}^{=|x|^s=q^{-ks}} \, dx^{\times} \\
&= \sum_{k=n_\psi}^{+\infty} q^{-k s} \int_{\varpi^k \OO^{\times}} \, dx^{\times} \\
&= \frac{q^{-n_\psi s}}{1-q^{-s}} \operatorname{Vol}\left(\OO^{\times}, dx^{\times}\right) .
\end{aligned}
$$
\end{proof}

\begin{lemme}~
\begin{aligned}[t]
    \text{Si } n_\chi=0 \quad  \hat{g}&= | \OO / \mathcal{D} |^{-1/2} q^{-n_\psi} \1_{\OO}, \\
    \text{Si } n_\chi>0  \quad \hat{g}&= | \OO / \mathcal{D} |^{-1/2} q^{n_\chi-n_\psi} \1_{-U_{n_\chi}}.
\end{aligned}
\end{lemme}


\begin{proof}~
$\hat{g}(y)=\int_F g(x) \psi(x y) d x=\int_{\varpi^{n_\psi-n_\chi} \OO} \psi(x) \psi(x y) d x$
$$
=\int_{\varpi^{n_\psi-n_\chi} \OO} \psi(x(y+1)) d x
$$
Or $\operatorname{Cond}(\psi)=\varpi^{n_\psi} \OO$ et $\operatorname{Cond}\left(\psi_{y+1}\right)=(y+1)^{-1} \varpi^{n_\psi} \OO$. \\
L'intégrale est nulle sauf si $\varpi^{n_\psi-n_\chi} \OO \subseteq (y+1)^{-1} \varpi^{n_\psi} \OO$, c'est-à-dire, sauf si $y+1 \in \varpi^{n_\chi} \OO$, c'est-à-dire sauf si $y \in -U_{n_\chi}$. \\

Ainsi \begin{aligned}[t]
\hat{g}(y) & =\1_{-U_{n_\chi}}(y) \cdot \operatorname{Vol}\left(\varpi^{n_\psi-n_\chi} \OO, dx\right) \\
& =\1_{-U_{n_\chi}}(y) \cdot|\varpi|^{n_\psi-n_\chi} \quad \operatorname{Vol}(\OO, dx) \\
& =\1_{-U_{n_\chi}}(y) \cdot q^{n_\chi-n_\psi} \cdot| \OO / \mathcal{D} |^{-1/2} 
\end{aligned}
\ \\
\qedhere
\end{proof}

