Zeros of quadratic Dirichlet $L$-functions

Source abstract

We consider Dirichlet $L$-functions $L(s, χ)$ where $χ$ is a quadratic character to modulus $q$. We show that for $χ$ even (resp. odd) $L(σ, χ)\neq 0$ for all $σ>0$ provided that $q\leq 10^{9}$ (resp. $q\leq 10^{10})$. In particular, this result rules out Siegel--Landau zeros for such $L$-functions. Combined, these results improve on previous work by nearly four orders of magnitude. We also list some applications of this work.