  We study the $1$-level density of low-lying zeros of Dirichlet $L$-functions
in the family of all characters modulo $q$, with $Q/2 < q\leq Q$. For test
functions whose Fourier transform is supported in $(-3/2, 3/2)$, we calculate
this quantity beyond the square-root cancellation expansion arising from the
$L$-function Ratios Conjecture of Conrey, Farmer and Zirnbauer. We discover the
existence of a new lower-order term which is not predicted by this powerful
conjecture. This is the first family where the 1-level density is determined
well enough to see a term which is not predicted by the Ratios Conjecture, and
proves that the exponent of the error term $Q^{-\frac 12 +\epsilon}$ in the
Ratios Conjecture is best possible. We also give more precise results when the
support of the Fourier Transform of the test function is restricted to the
interval $[-1,1]$. Finally we show how natural conjectures on the distribution
of primes in arithmetic progressions allow one to extend the support. The most
powerful conjecture is Montgomery's, which implies that the Ratios Conjecture's
prediction holds for any finite support up to an error $Q^{-\frac 12
+\epsilon}$.
