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 $\sigma<-1/2$.