Extreme values of Hecke $L$-functions to angular characters

Let $K$ be an imaginary quadratic number field and let $L(s,{\xi }_{\mathrm{\ell }})$ denote the Hecke $L$-function to an angular character ${\xi }_{\mathrm{\ell }}$ with frequency $\mathrm{\ell }$. We detect values of $\log \mathrm{\mid }L(\frac{1}{2},{\xi }_{\mathrm{\ell }})\mathrm{\mid }$ with size at least $(\sqrt{2}+o_K(1))(\log X\mathrm{/}\log \log X{)}^{1\mathrm{/}2}$ along each dyadic range $X⩽\mathrm{\ell }⩽2X$. This result relies on the resonance method, which is applied for the first time to this family of $L$-functions, where the classification and extraction of diagonal terms depends on the geometry of the complex embedding of $K$.