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\newtheorem{thm}{Theorem}

\begin{document}
In order to state the main theorem of this section we need to introduce a bit more notation. We write $\alpha_S(s,\chi)$ for the Siegel series attached to the symmetric matrix $S$ and to the character $\chi$, as defined for example in \cite[Chapter III]{Sh96}. Moreover, by \cite[Theorem 13.6]{Sh96}, we have
\begin{equation}\label{Siegel_series}
\alpha_S (s,\chi) = \left(L(s,\chi) \prod_{i=1}^{[l/2]} L(2s-2i,\chi^{2})\right)^{-1} g_S(s,\chi) 
\end{equation}
for some analytic function $g_S(s,\chi)$ of the form $g_S(s,\chi) = G(\chi(\pi)q^{-s})$ for some polynomial $G(X) \in \mathbb{Z}[X]$ of constant term one. Moreover if $S$ is regular, that is, $\det(2S) = \mathfrak{o}^{\times}$ for $l$ even and $\det(2S) = 2\mathfrak{o}^{\times}$ for $l$ odd, then $g_S(s,\chi) = 1$. \newline

The following theorem generalizes a result due to Murase and Sugano \cite{MS}, where the case of $l=1$ and $\chi$ trivial is considered.

\begin{thm}\label{good Euler factor}
With the notation as above,
$$L(\xi,\chi,s) = \frac{g_S(s+n+l/2,\chi)}{g_S(s+l/2,\chi)} \Lambda(\chi,s)\int_{\bo{Z}\setminus \bo{G}}  \nu_{\chi,s+n+l/2}(\bo{g})\phi_{\xi}(\bo{g})d\bo{g}\Lambda(\chi,s),$$
where
$$\Lambda(\chi,s):=\begin{cases}  \prod_{i=1}^{n} L(2s+2n-2i,\chi^{2}) &\mbox{if } l \in 2\mathbb{Z},  \\ 
\prod_{i=1}^{n} L(2s+2n-2i+1,\chi^{2})  & \mbox{if }  l \not \in 2\mathbb{Z}.\end{cases} $$
In particular,
$$L(\xi,\chi,s) =  B(\xi,\chi,s+n+l/2) \frac{g_S(s+n+l/2,\chi)}{g_S(s+l/2,\chi)}\Lambda(\chi,s).$$
\end{thm}

The rest of this subsection is devoted to a proof of Theorem \ref{good Euler factor}. First we extend some calculations of Murase and Sugano \cite{MS}. Denote by $\sigma_{n_1,n_2}$ the characteristic function of $M_{n_1,n_2}(\mathfrak{o})$ and let
\begin{multline*}
F(s,\chi,\bo{g}) := F(s,\chi, h g ) :=\\
\int_{GL_{2n+l}(F_v)} \sigma_{2n+l,4n+2l} \left( \left( y \left(\begin{matrix} 1_l & 0 \\ 0 & g \end{matrix} \right), y \alpha(h) \right) \right) \chi(\det(y)) |\det(y)|^{s+n + l/2} d^*y, 
\end{multline*}
where for $h = (\lambda,\mu,\kappa) \in H$ we set
$$
\alpha(h) := \left( \begin{matrix}   \kappa - \lambda \transpose{\mu} & - \lambda & -\mu \\                           
\transpose{\mu} & 1_n & 0 \\
\transpose{\lambda} & 0 & 1_n  \end{matrix} \right).
$$
Define also
$$
\mathcal{F}(s,\chi,\bo{g}) := \int_{\mathcal{Z}} F(s,\chi,(0,0,\kappa) \bo{g}) \psi_S(\kappa) d \kappa .
$$

\begin{thebibliography}{10}
\bibitem{MS} A. Murase and T. Sugano, Whittaker-Shintani functions on the symplectic group of Fourier-Jacobi type, {\em Compositio Mathematica}, 79 (1991), 321-349.
%
\bibitem{Sh96} G. Shimura, Euler Products and Eisenstein Series, Conference Board of the Mathematical Sciences (CBMS), Number 93, AMS, 1996.
\end{thebibliography}
\end{document}