On the effective non-vanishing of Hecke–Maass L -functions at special points

Abstract

Abstract In this article, we consider the non-vanishing problem for the family of special Hecke–Maass L -values L ( 1 / 2 + i t f , f ) $ L (1/2+it_f, f) $ upper L left parenthesis 1 divided by 2 plus i t Subscript f Baseline comma f right parenthesis with f ( z ) $f (z)$ f left parenthesis z right parenthesis in an orthonormal basis of (even or odd) Hecke–Maass cusp forms of Laplace eigenvalue 1 / 4 + t f 2 $1/4 + t_f^2$ 1 divided by 4 plus t Subscript f Superscript 2 ( t f > 0 $t_f> 0$ t Subscript f Baseline greater than 0 ). We prove that 20% of L ( 1 / 2 + i t f , f ) $L (1/2+it_f, f)$ upper L left parenthesis 1 divided by 2 plus i t Subscript f Baseline comma f right parenthesis for t f ⩽ T $ t_f \leqslant T$ t Subscript f Baseline less than or slanted equals upper T do not vanish as T → ∞ $T \rightarrow \infty $