
\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}



Preuve $=\operatorname{Pour} \operatorname{Re}(s)>0$, on a: termes nuls si $k<n_\psi \quad=|x|^s=q^{-k s}$
$$
\begin{aligned}
Z(g, X)=\int_{F^x} g(x) X(x) \, dx^x & =\sum_{k=-\infty}^{+\infty} \int_{\pi^k O^x} \mathbb{1}_{\pi^{m_\psi} O}(x) \psi(x) X(x) \, dx^x \\
& =\sum_{k=n_\psi}^{+\infty} q^{-k s} \int_{\sigma^k O^x} \, dx^x=\frac{9^{-n_\psi}}{1-q^{-5} \mathcal{L}} \operatorname{Vol}\left(O^x, d x^x\right)
\end{aligned}
$$

- LEM: Si $n_x=0$, $\hat{g}=10 /\left.\mathscr{D}\right|^{-1 / 2} 9^{-n_\psi} 1_0$
Si $n_x>0$, $\hat{g}=|0 / /|^{-1 / 2} q^{n_x-n_\psi} 1_{-u_{n_x}}$
