Non-vanishing of $L$-functions over function fields

I will talk about some results concerning the non-vanishing of $L$-functions associated
to fixed order characters $\ell$ at the central point over functions fields. More specifically,
I will explain how one can compute the one-level density of zeros in a thin family of such
$L$-functions and obtain a positive proportion of non-vanishing for any $\ell$ which goes
to $0$ as $\ell$ goes to infinity. This is based on joint work with C. David and M. Lalin.