Non-vanishing of Maass form $L$-functions at the central point

Abstract

In this paper, we consider the family {Lj(s)}∞j=1 of L-functions associated to an orthonormal basis {uj}∞j=1 of even Hecke-Maass forms for the modular group SL(2,Z) with eigenvalues {λj = κ2j + 1/4}∞j=1. We prove the following effective non-vanishing result: At least 50% of the central values Lj(1/2) with κj ≤ T do not vanish as T → ∞. Furthermore, we establish effective non-vanishing results in short intervals.