Asymptotics for twisted first and second moments of r-th order Hecke L-functions yield a positive proportion of non-vanishing central values for square-free and r-th power-free ideal families when r≥3.
Twisted second moment of primitive cubic L-functions
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abstract
We investigate the mean value of the twisted second moment of primitive cubic $L$-functions over $\mathbb{F}_q(T)$ in the non-Kummer setting. Specifically, we study the sum \begin{equation*} \sum_{\substack{\chi\ primitive\ cubic\\ genus(\chi)=g}}\chi(h_1)\bar{\chi}(h_2)|L_q(\frac{1}{2}, \chi)|^2, \end{equation*} where $L_q(s,\chi)$ denotes the $L$-function associated with primitive cubic character $\chi$. Employing a double Dirichlet series approach, we establish an error term of size
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On the second moment and non-vanishing of central values of Hecke $L$-functions of $r$-th order characters
Asymptotics for twisted first and second moments of r-th order Hecke L-functions yield a positive proportion of non-vanishing central values for square-free and r-th power-free ideal families when r≥3.