Higher-order Hecke L-function moment conjecture with switching behavior

Let $\mathcal{F}_\ell$ be a family of Hecke characters of order $\ell$ with size $Q$, let $c_\chi$ be the conductor of $\chi$, and let $w:(0,\infty)\to\mathbb{R}$ be smooth with compact support in $(0,1)$. For $\ell\geq4$ and $s\in\mathbb{C}$, define the weighted first moment by

$$M_\ell(s,Q)=\sum_{\chi\in\mathcal{F}_\ell}L(s,\chi)w\left(\frac{N(c_\chi)}{Q}\right).$$

Higher-order switching conjecture.

$$M_\ell(s,Q)=\begin{cases}C_{\ell,s,w}Q+D_{\ell,s,w}Q^{1+\frac1\ell-s}(1+o(1))&\frac1\ell<\operatorname{Re}(s)\leq1,\\\\C_{\ell,w}Q\log Q+D_{\ell,w}Q(1+o(1))&\operatorname{Re}(s)=\frac1\ell,\\\\D_{\ell,s,w}Q^{1+\frac1\ell-s}+C_{\ell,s,w}Q(1+o(1))&0\leq\operatorname{Re}(s)<\frac1\ell. \end{cases}$$

Here the constants are those appearing in the predicted asymptotic. This refines the David–Meisner conjecture by predicting secondary terms and the switching of the main and secondary terms across $\operatorname{Re}(s)=1/\ell$. The source motivates this conjecture from the proved quadratic and cubic cases, while the higher-order case $\ell\geq4$ remains open.