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 \(\mathcal {S}_k(N,\chi )\).