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 >0 and sufficiently large X, there are X^1- fundamental discriminants 8d with X < d 2X and (d, 2q) =1 such that |L (1/2, F ₈₃) | is large.