On the second moment and non-vanishing of central values of Hecke $L$-functions of $r$-th order characters

In this paper, we establish asymptotic formulas for the first and second twisted moments of $r$-th order Hecke $L$-functions over global fields that contain the $2r$-th roots of unity, for $r\ge 3$.

We focus primarily on algebraic number fields.

As a consequence, we establish a positive proportion of non-vanishing central values for these $L$-functions, specifically for families of both square-free and $r$-th power-free ideals.

Our approach is based on the machinery of multiple Dirichlet series.