Non-vanishing of Maass form L-functions at the critical point

Abstract

In this paper, we consider the family $\{L_j(s){\}}_{j=1}^{\infty }$ of $L$-functions associated to an orthonormal basis $\{u_j{\}}_{j=1}^{\infty }$ of even Hecke-Maass forms for the modular group $SL(2,Z)$ with eigenvalues $\{{\lambda }_j={\kappa }_j^2+1/4{\}}_{j=1}^{\infty }$. We prove the following effective non-vanishing result: At least $50\%$ of the central values $L_j(1/2)$ with ${\kappa }_j\le T$ do not vanish as $T\to \infty$. Furthermore, we establish effective non-vanishing results in short intervals.

improved results; a new co-author added