  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 $X < d \leq 2X$ and ${(d, 2q)=1}$ such that ${|L(1/2, F
\otimes \chi_{8d})| }$ is large.
