  We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions
attached to real primitive characters of conductor at most $X$. Under the
Generalized Riemann Hypothesis, we give an asymptotic expansion of this
quantity in descending powers of $\log X$, which is valid when the support of
the Fourier transform of the corresponding even test function $\phi$ is
contained in $(-2,2)$. We uncover a phase transition when the supremum $\sigma$
of the support of $\hat \phi$ reaches $1$, both in the main term and in the
lower order terms. A new lower order term appearing at $\sigma=1$ involves the
quantity $\hat \phi (1)$, and is analogous to a lower order term which was
isolated by Rudnick in the function field case.
