A hybrid asymptotic formula for the second moment of Rankin-Selberg L -functions

Abstract

Let g be a fixed modular form of full level, and let {fj, k} be a basis of holomorphic cuspidal newforms of even weight k, fixed level and fixed primitive nebentypus. We consider the Rankin–Selberg L-functions L(½+it, fj, k ⊗ g) and compute their second moment over t ≈ T and k ≈ K. For K3/4+ε⩽ T ⩽ K5/4−ε, we obtain an asymptotic formula with a power-saving error term. Our result covers the second moment of L(½+it+ir, fj, k)L(½+it−ir, fj, k) for any fixed real number r, hence also the fourth moment of L(½+it, fj, k). For the proof, we develop a precise uniform approximate functional equation with explicit dependence on the archimedean parameters.