Extreme values of Dirichlet type $L$-functions

In this talk, we will report on a joint work with Sanoli Gun, where we study extreme values of $L$-functions attached to quadratic twists of Dirichlet characters. We show that for any $\epsilon >0$ and Dirichlet character $F$ of odd conductor $q$, not necessarily a primitive form, there exists at least $X^{1-\epsilon}$ fundamental discriminants $8d$ with $X< d \le 2X$ and $(d, 2q) =1$ such that $|L(1/2, F \otimes \chi_{8d})|$ takes large values.