  We give a geometric criterion for Dirichlet $L$-functions associated to
cyclic characters over the rational function field $\mathbb{F}_q(t)$ to vanish
at the central point $s=1/2$. The idea is based on the observation that
vanishing at the central point can be interpreted as the existence of a map
from the projective curve associated to the character to some abelian variety
over $\mathbb{F}_q$. Using this geometric criterion, we obtain a lower bound on
the number of cubic characters over $\mathbb{F}_q(t)$ whose $L$-functions
vanish at the central point where $q=p^{4n}$ for any rational prime $p \equiv 2
\bmod 3$. We also use recent results about the existence of supersingular
superelliptic curves to deduce consequences for the $L$-functions of Dirichlet
characters of other orders.
