  Let $\psi$ be a real primitive character modulo $D$. If the $L$-function
$L(s,\psi)$ has a real zero close to $s=1$, known as a Landau-Siegel zero, then
we say the character $\psi$ is exceptional. Under the hypothesis that such
exceptional characters exist, we prove that at least fifty percent of the
central values $L(1/2,\chi)$ of the Dirichlet $L$-functions $L(s,\chi)$ are
nonzero, where $\chi$ ranges over primitive characters modulo $q$ and $q$ is a
large prime of size $D^{O(1)}$. Under the same hypothesis we also show that,
for almost all $\chi$, the function $L(s,\chi)$ has at most a simple zero at $s
= 1/2$.
