We prove a functional equation for a vector valued real analytic Eisenstein series transforming with the Weil representation of $\operatorname{Sp}(n,\mathbb{Z})$ on $\mathbb{C}[(L^{\prime }/L)^{n}]$. By relating such an Eisenstein series with a real analytic Jacobi Eisenstein series of degree $n$, a functional equation for such an Eisenstein series is proved. Employing a doubling method for Jacobi forms of higher degree established by Arakawa, we transfer the aforementioned functional equation to a zeta function defined by the eigenvalues of a Jacobi eigenform. Finally, we obtain the analytic continuation and a functional equation of the standard $L$-function attached to a Jacobi eigenform, which was already proved by Murase, however in a different way.

中文翻译：

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爱森斯坦级数和标准函数的解析性质

我们证明了向量值实解析 Eisenstein 级数转换的函数方程，其 Weil 表示为$\operatorname{Sp}(n,\mathbb{Z})$在$\mathbb{C}[(L^{\prime }/L)^{n}]$. 通过将这样的 Eisenstein 级数与真实的解析 Jacobi Eisenstein 度数级数联系起来$n$，证明了这种 Eisenstein 级数的函数方程。采用由 Arakawa 建立的高阶 Jacobi 形式的加倍方法，我们将上述函数方程转换为由 Jacobi 特征型的特征值定义的 zeta 函数。最后，我们得到了标准的解析延拓和函数方程$L$- 附加到 Jacobi 特征形式的函数，这已经被 Murase 证明了，但是以不同的方式。