 
\begin{proof}~

Si $n_\chi>0$, on a pour $\Re(s)<1$ : \\
$\mathcal{Z}\left(\hat{g}, \chi^\vee\right)=\int_{F^*} \hat{g}(x) \chi^\vee(x) dx^\times=| \OO / \mathcal{D} |^{-1/2} q^{n_\chi-n_\psi} \underbrace{\int_{-U_{n_\chi}} \chi^\vee(x) dx^\times}_{ = \chi^\vee (-1) \operatorname{Vol}(U_{n_\chi},dx^\times)} $. \\

\ \\
Or $\eta(-1)= \pm 1$ et $\chi^\vee(-1)=\bar{\eta(-1)}=\eta(-1) = \chi(-1)$, et: 

$\operatorname{Vol}\left(U_{n_\chi}, dx^\times\right)=\left((q-1) q^{n_\chi-1}\right)^{-1} \operatorname{Vol}\left(\OO^\times, dx^\times\right)$.\\
D'où $\mathcal{Z}\left(\hat{g}, \chi^\vee\right)=| \OO / \mathcal{D} |^{-1} q^{-n_\psi } \frac{\chi(-1)}{1-q^{-1}}$.\\

Si $n_\chi=0 :\begin{aligned}[t]
\mathcal{Z}\left(\hat{g}, \chi^\vee\right)&=\int_{F^*}| \OO / \mathcal{D} |^{-1/2} q^{-n_\psi}  \1_{\OO}(x) \chi^\vee(x) dx^\times \\
& =| \OO / \mathcal{D} |^{-1/2} q^{-n_\psi} \sum_{k=0}^{+\infty} q^{-k(1-s)} \operatorname{Vol}\left(\O\OO^\times, dx^\times\right)=| \OO / \mathcal{D} |^{-1} \frac{q^{-n_\psi}}{1-q^{s-1}} \\
&
\end{aligned}
\end{proof}


Preuve: Si $m_x>0$, on a pour $\operatorname{Re}(s)<1$ : $\quad=X^2(-1) V_0\left(U_{m_x} d x^x\right)$
$$
Z\left(\hat{g}, X^v\right)=\int_{F^x} \hat{g}(x) X^v(x) \, dx^x=10 /\left.\mathscr{D}\right|^{-1 / 2} q^{n_x-n_\psi} \int_{-U_{n_x}} X^v(x) \, dx^x
$$
or $\eta(-1)= \pm 1$ et $x^2(-1)=\overline{\eta(-1)}=\overline{\eta(-1)}$, et :
$\operatorname{Vol}\left(U_{n_x}, d x^x\right)=\left(\left(q^{+}+1\right) q^{n_x-1}\right)^{-1} \operatorname{Vol}\left(\mathbb{O}^x, d x^x\right)$

D'où $Z\left(\hat{g}, x^v\right)=|\mathbb{O} / \theta|^{-1} q^{-m_\psi} \frac{x(-1)}{1-q^{-1}}$.
$$
\begin{aligned}
& \text { Si } m_x=0: Z\left(\hat{g}, X^v\right)=\int_{F^x}|0 / \phi|^{-1 / 2} q^{-n_\psi} \mathbb{1}_0(x) X^\nu(x) \, dx^x \\
& =10 /\left.\mathscr{D}\right|^{-1 / 2} q^{-n} \sum_{k=0}^{+\infty} q^{-k(1-s)} \operatorname{Vol}\left(\mathbb{O}^x, d c^x\right)=10 / /\left.\right|^{-1} \frac{q^{-n \psi}}{1-q^{s-1}} \\
&
\end{aligned}
$$
