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$L$-functions: lower order terms for extended support
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 
$\unicode[STIX]{x1D719}$ is contained in 
$(-2,2)$. We uncover a phase transition when the supremum 
$\unicode[STIX]{x1D70E}$ of the support of 
$\widehat{\unicode[STIX]{x1D719}}$ reaches 
$1$, both in the main term and in the lower order terms. A new lower order term appearing at 
$\unicode[STIX]{x1D70E}=1$ involves the quantity 
$\widehat{\unicode[STIX]{x1D719}}(1)$, and is analogous to a lower order term which was isolated by Rudnick in the function field case.
