Repulsive behavior in an exceptional family

TL;DR

This paper investigates how a Landau-Siegel zero affects the distribution of zeros in quadratic twist families, revealing a resonance phenomenon and providing explicit lower order terms for zero distribution.

Contribution

It introduces explicit lower order terms for the 1-level density in quadratic twist families, illustrating the resonance effect of Landau-Siegel zeros on zero distribution.

Findings

Landau-Siegel zero causes a resonance in zero distribution

Explicit lower order terms describe zero distribution effects

Demonstrates the Deuring-Heilbronn phenomenon in this context

Abstract

The existence of a Landau-Siegel zero leads to the Deuring-Heilbronn phenomenon, here appearing in the 1-level density in a family of quadratic twists of a fixed genus character L-function. We obtain explicit lower order terms describing the vertical distribution of the zeros, and realize the influence of the Landau-Siegel zero as a resonance phenomenon.