Root numbers for twisted Fermat quotient curves

Let $\mathrm{\ell }$ be an odd prime, $N\ge 1$ be an integer, and $\delta \ge 1$ be a ${\mathrm{\ell }}^N$-th power free integer such that ${\mathrm{o}\mathrm{r}\mathrm{d}}_{\mathrm{\ell }}(\delta )=0$ or $\mathrm{\ell }\nmid {\mathrm{o}\mathrm{r}\mathrm{d}}_{\mathrm{\ell }}(\delta )$. In this paper, we give an explicit formula for the root number of the Hecke character associated with a certain quotient curve of the twisted Fermat curve $X^{{\mathrm{\ell }}^N}+Y^{{\mathrm{\ell }}^N}=\delta$. This result gives a generalization of Stoll (2002) and Shu (2021).