David–Meisner conjecture for special values of higher-order Hecke L-functions

Let $K$ be a number field containing the $\ell$-th roots of unity, let $\mathcal{O}_K$ be its ring of integers, and let $\mathcal{F}_\ell$ be a family of Hecke characters of order $\ell$ attached to $\ell$-th order residue symbols, of size $Q\geq2$. For each character $\chi$, let $c_\chi$ be its conductor, and let $w:(0,\infty)\to\mathbb{R}$ be smooth with compact support in $(0,1)$. David–Meisner conjecture. For $\ell\geq3$ and $s\in(0,1)$,

$$\sum_{\chi\in\mathcal{F}_\ell}L(s,\chi)w\left(\frac{N(c_\chi)}{Q}\right)\sim\begin{cases}C_{\ell,s,w}Q&\frac1\ell<s<1,\\\\C_{\ell,s,w}Q\log Q&s=\frac1\ell, \end{cases}$$

where $C_{\ell,s,w}$ depends on $\ell$, $s$, and $w$. This is the number-field translation of the function-field conjecture of David and Meisner; the source notes that the cubic case is proved in the relevant function-field setting and that the present paper proves the cubic number-field analogue.