
\begin{lemme}
    On a $\varepsilon(\chi, \psi) \varepsilon\left(\chi^\vee, \psi\right)=\chi(-1)$.
\end{lemme}


\begin{proof}
    $\mathcal{Z}(\hat{\hat{f}}, \chi) \stackrel{\text { Fourier }}{=} \int_{F^*} f(-x) \chi(x) d x^\times=\chi(-1) \mathcal{Z}(f, \chi)$. \\
    Or $\frac{\mathcal{Z}\hat{\hat{f}}, \chi)}{L(\chi)}=\varepsilon\left(\chi^\vee, \psi\right) \frac{\mathcal{Z}\left(\hat{f}, \chi^\vee\right)}{L\left(\chi^\vee\right)}=\varepsilon\left(\chi^\vee, \psi\right) \varepsilon(\chi, \psi) \frac{\mathcal{Z}(f, \chi)}{L(\chi)}$.
\end{proof}


\begin{lemme}
    On a $\varepsilon(\chi, \psi) \varepsilon\left(\chi^\vee, \psi\right)=\chi(-1)$.
\end{lemme}


\begin{proof}
    $\mathcal{Z}(\hat{\hat{f}}, \chi) \stackrel{\text { Fourier }}{=} \int_{F^*} f(-x) \chi(x) d x^\times=\chi(-1) \mathcal{Z}(f, \chi)$. \\
    Or $\frac{\mathcal{Z}\hat{\hat{f}}, \chi)}{L(\chi)}=\varepsilon\left(\chi^\vee, \psi\right) \frac{\mathcal{Z}\left(\hat{f}, \chi^\vee\right)}{L\left(\chi^\vee\right)}=\varepsilon\left(\chi^\vee, \psi\right) \varepsilon(\chi, \psi) \frac{\mathcal{Z}(f, \chi)}{L(\chi)}$.
\end{proof}
