On extreme values of quadratic twists of Dirichlet-type $L$-functions

In a recent work arXiv:2004.14450, it has been shown that $L$-functions associated with arbitrary non-zero cusp forms take large values at the central critical point. The goal of this note is to derive analogous results for twists of Dirichlet-type functions. More precisely, for an odd integer $q>1$, let $F$ be a non-zero $\mathbb{C}$-linear combination of primitive, complex, even Dirichlet characters of conductor $q$. We show that for any $\epsilon >0$ and sufficiently large $X$, there are $\gg X^{1-\epsilon }$ fundamental discriminants $8d$ with $X<d\le 2X$ and $(d,2q)=1$ such that $\mathrm{\mid }L(1\mathrm{/}2,F\otimes {\chi }_{8d})\mathrm{\mid }$ is large.