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

We prove a one-sided central limit theorem for $\log \mathrm{\mid }L(\frac{1}{2},\chi )\mathrm{\mid }$ in a thin family of $n$-th order Hecke characters, for $n\ge 3$. The proof combines the method of Radziwiłł and Soundararajan with a recent asymptotic formula of Diaconu, Ion, Paşol, 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 $0<k<1$.