  Katz and Sarnak conjectured that the statistics of low-lying zeros of various
family of $L$-functions matched with the scaling limit of eigenvalues from the
random matrix theory. In this paper we confirm this statistic for a family of
primitive Dirichlet $L$-functions matches up with corresponding statistic in
the random unitary ensemble, in a range that includes the off-diagonal
contribution. To estimate the $n$-level density of zeros of the $L$-functions,
we use the asymptotic large sieve method developed by Conrey, Iwaniec and
Soundararajan. For the random matrix side, a formula from Conrey and Snaith
allows us to solve the matchup problem.
