  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 $\log |L(\tfrac12,\xi_{\ell})|$ with size
at least $(\sqrt{2} + o_K(1))(\log X / \log \log X)^{1/2}$ along each dyadic
range $X \leqslant \ell \leqslant 2X$. 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 $K$.
