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

Abstract

Let $K$ be an imaginary quadratic number field and let $L(s,{\xi }_{\ell })$ denote the Hecke $L$-function to an angular character ${\xi }_{\ell }$ with frequency $\ell$. We detect values of $\log |L(\frac{1}{2},{\xi }_{\ell })|$ with size at least $(\sqrt{2}+o_K(1))(\log X/\log \log X{)}^{1/2}$ along each dyadic range $X⩽\ell ⩽2X$. 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 $K$.