The highest lowest zero of general L-functions

TL;DR

This paper investigates the distribution of zeros of general L-functions, providing counterexamples to previous hypotheses and establishing new bounds under certain conjectural conditions.

Contribution

It presents a counterexample to Miller's zero interval for general L-functions and proposes a broader zero interval under additional conjectural assumptions.

Findings

Counterexample with first zero at t≈14.496

New zero interval with t≈22.661 under conjectural conditions

Shows Miller's result does not extend to all 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 $\frac12+i t$ 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 $t_2\approx 22.661$.