Counting zeros of Dirichlet $L$-functions

We give explicit upper and lower bounds for $N(T,\chi )$, the number of zeros of a Dirichlet $L$-function with character $\chi$ and height at most $T$. Suppose that $\chi$ has conductor $q>1$, and that $T\geq 5/7$. 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. \end{equation*} We give slightly stronger results for small $q$ and $T$. Along the way, we prove a new bound on $|L(s,\chi )|$ for $\sigma <-1/2$.