  Y\i ld\i r\i m has classified zeros of the derivatives of Dirichlet
$L$-functions into trivial zeros, nontrivial zeros and vagrant zeros. In this
paper we remove the possibility of vagrant zeros for $L'(s,\chi)$ when the
conductors are large to some extent. Then we improve asymptotic formulas for
the number of zeros of $L'(s,\chi)$ in
$\{s\in\mathbb{C}:\operatorname{Re}(s)>0, |\operatorname{Im}(s)|\leq T\}$. We
also establish analogues of Speiser's theorem, which characterize the
generalized Riemann hypothesis for $L(s,\chi)$ in terms of zeros of
$L'(s,\chi)$, when the conductor is large.

Number Theory