The asymptotic formula for the averages of the product of twisted L-functions

Abstract

Suppose that q and r are two distinct large primes. Let g be a cuspidal Hecke eigenform of level 1 and even weight \(k_1\) and \(H_{k_2}(q)\) be the set of cuspidal Hecke eigenforms of level q and even weight \(k_2\). In this paper, we investigate the averages of the product of the central values of two L-functions of modular forms \(f\in H_{k_2}(q)\) and g twisted by all primitive Dirichlet characters modulo r and obtain an asymptotic formula with a power saving error term under certain assumptions.