Conjectures and Experiments Concerning the Moments of $L (1/2, \chi_d)$

Abstract

We report on some extensive computations and experiments concerning the moments of quadratic Dirichlet $L$-functions at the critical point. We computed the values of $L (1/2, \chi_d)$ for $−5 × 10^{10} \lt d \lt 1.3 × 10^{10}$ in order to numerically test conjectures concerning the moments $\sum_{|d|\lt X} L (1/2, \chi_d)^k$. Specifically, we tested the full asymptotics for the moments conjectured by Conrey, Farmer, Keating, Rubinstein, and Snaith, as well as the conjectures of Diaconu, Goldfeld, Hoffstein, and Zhang concerning additional lower-order terms in the moments. We also describe the algorithms used for this large-scale computation.