Distribution of central values of Hecke $L$-functions of $n$-th order characters

Abstract

We prove a one-sided central limit theorem for $\log |L(\tfrac 12,\chi)|$ in a thin family of $n$-th order Hecke characters, for $n \geq 3$. The proof combines the method of Radziwi{\l}{\l} and Soundararajan with a recent asymptotic formula of Diaconu, Ion, Pa\c{s}ol, and Popa for the twisted second moment with a power-saving error term. We also obtain upper bounds of the conjectured order of magnitude for the $2k$-th moments for $0