An average of special values of Dirichlet series of Rankin–Selberg type

Abstract

For an integer n and a Dirichlet character $\xi$ modulo N, we denote by $\mathcal {S}_n(N,\xi )$ the space of cusp forms of weight n with respect to $\varGamma _0(N)$ and nebentypus $\xi$. Here $\varGamma _0(N)$ is the Hecke congruence subgroup. Let k, l be nonnegative integers with $k-l \ge 2$ and $\chi ,\psi$ Dirichlet characters modulo N. For a fixed $g \in \mathcal {S}_l(N,\psi )$, we give an explicit expression for the average of special values of the Dirichlet series $D(s,f \otimes g)$ of Rankin–Selberg type at each $s=m \in {\mathbb {Z}}$ with $\frac{k+l}{2}-1< m<k$ as f ranges over an orthogonal basis of $\mathcal {S}_k(N,\chi )$.