Zeros of the first derivative of Dirichlet L-functions

Yıldırım 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,χ) when the conductors are large to some extent. Then we improve asymptotic formulas for the number of zeros of L′(s,χ) in {s∈C:Re(s)>0,|Im(s)|≤T}. We also establish analogues of Speiser's theorem, which characterize the generalized Riemann hypothesis for L(s,χ) in terms of zeros of L′(s,χ), when the conductor is large.