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 C-linear combination of primitive, complex, even Dirichlet characters of conductor q. We show that for any ε>0 and sufficiently large X, there are X1-ε fundamental discriminants $8d$ with X < d ≤ 2X and (d, 2q)=1 such that |L(1/2, F ⊗ χ8d)| is large.