  We consider Dirichlet $L$-functions $L(s, \chi)$ where $\chi$ is a real,
non-principal character modulo $q$. Using Pintz's refinement of Page's theorem,
we prove that for $q\geq 3$ the function $L(s, \chi)$ has at most one real zero
$\beta$ with $1- 1.011/\log q < \beta < 1$.
