The highest lowest zero of general L-functions

Abstract

Stephen D. Miller showed that, assuming the generalized Riemann Hypothesis, every entire $L$-function of real archimedian type has a zero in the interval $\frac{1}{2}+it$ with $-t_0<t<t_0$, where $t_0\approx 14.13$ corresponds to the first zero of the Riemann zeta function. We give an example of a self-dual degree-4 $L$-function whose first positive imaginary zero is at $t_1\approx 14.496$. In particular, Miller's result does not hold for general $L$-functions. We show that all $L$-functions satisfying some additional (conjecturally true) conditions have a zero in the interval $(-t_2,t_2)$ with $t_2\approx 22.661$.

Added higher precision values for coefficients. Final version, to appear in JNT

References in corpus (1)

• Maass forms on GL(3) and GL(4)

Cited by in corpus (1)

• Maass forms on GL(3) and GL(4)