Koecher–Maass series of the Ikeda lift for U ( m , m )

Abstract

Let $K=\mathbf{Q}\left(\sqrt{-D}\right)$ be an imaginary quadratic field with discriminant $-D$, and let $\chi$ be the Dirichlet character corresponding to the extension $K/\mathbf{Q}$. Let $m=2n$ or $2n+1$ with $n$ a positive integer. Let $f$ be a primitive form of weight $2k+1$ and character $\chi$ for ${\Gamma }_0\left(D\right)$ or a primitive form of weight $2k$ for ${SL}_2\left(\mathbf{Z}\right)$ according to whether $m=2n$ or $m=2n+1$. For such an $f$ let $I_m\left(f\right)$ be the lift of $f$ to the space of Hermitian modular forms constructed by Ikeda. We then give an explicit formula of the Koecher–Maass series $L(s,I_m(f\left)\right)$ of $I_m\left(f\right)$. This is a generalization of Mizuno.