The Quadratic Character Experiment

Abstract

A fast new algorithm is used compute the zeros of $10^6$ quadratic character $L$-functions for negative fundamental discriminants with absolute value $d>10^{12}$. These are compared to the $1$-level density, including various lower-order terms. These terms come from, on the one hand, the explicit formula, and on the other, the $L$-functions ratios conjecture. The latter give a much better fit to the data, providing numerical evidence for the conjecture.