On the Effective Non-vanishing of Hecke--Maass $L$-functions at Special Points

Abstract

In this paper, we consider the non-vanishing problem for the family of special Hecke--Maass $L$-values $L(1/2+i{t}_f,f)$ with $f(z)$ in an orthonormal basis of (even or odd) Hecke--Maass cusp forms of Laplace eigenvalue $1/4+t_f^2$ ($t_f>0$). We prove that 20% of $L(1/2+i{t}_f,f)$ for $t_f⩽T$ do not vanish as $T\to \infty$. For comparison, it is known that the non-vanishing proportion is at least 25% for the central $L$-values $L(1/2,f)$. Further, 20% may be raised to 50% conditionally on the generalized Riemann hypothesis. Moreover, we prove non-vanishing results on the short interval $|t_f-T|⩽T^{\mu }$ for any $0<\mu <1$.

37 pages; many thanks to the referees for corrections; accepted by Canad. J. Math.. arXiv admin note: text overlap with arXiv:2506.08546