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

Surpassing the Ratios Conjecture in the 1-level density of Dirichlet $L$-functions