We give explicit upper and lower bounds for $N(T,\chi)$, the number of zeros a Dirichlet $L$-function with character $\chi$ height at most $T$. Suppose that has conductor $q>1$, $T\geq 5/7$. If $\ell=\log\frac{q(T+2)}{2\pi}> 1.567$, then \begin{equation*}
\left| N(T,\chi) - \left( \frac{T}{\pi} \log\frac{qT}{2\pi e} -\frac{\chi(-1)}{4}\right) \right|
\le 0.22737 \ell + 2 \log(1+\ell) 0.5.
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