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

Abstract

In this article, we consider the non-vanishing problem for the family of special Hecke–Maass L -values

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 $ upper T right arrow infinity

. For comparison, it is known that the non-vanishing proportion is at least 25% for the central L -values

L ( 1 / 2 , f ) $L (1/2, f)$ upper L left parenthesis 1 divided by 2 comma f right parenthesis

. 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 μ $|t_f-T| \leqslant T^{\mu }$ StartAbsoluteValue t Subscript f Baseline minus upper T EndAbsoluteValue less than or slanted equals upper T Superscript mu

for any

0 < μ < 1 $0 < \mu < 1$ 0 less than mu less than 1

.