In the Directed Steiner Network problem we are given an arc-weighted digraph G, a set of terminals T ⊆ V(G) with |T| = q, and an (unweighted) directed request graph R with V(R) = T. Our task is to output a subgraph H ⊆ G of the minimum cost such that there is a directed path from s to t in H for all st ∈ A(R). It is known that the problem can be solved in time |V(G)|O(|A(R)|) [Feldman&Ruhl, SIAM J. Comput. 2006] and cannot be solved in time |V(G)|o(|A(R)|) even if G is planar, unless the Exponential-Time Hypothesis (ETH) fails [Chitnis et al., SODA 2014]. However, the reduction (and other reductions showing hardness of the problem) only shows that the problem cannot be solved in time |V(G)|o(q), unless ETH fails. Therefore, there is a significant gap in the complexity with respect to q in the exponent. We show that Directed Steiner Network is solvable in time f(q) · |V(G)|O(cg·q), where cg is a constant depending solely on the genus of G and f is a computable function. We complement this result by showing that there is no f(q) · |V(G)|o(q2/log q) algorithm for any function f for the problem on general graphs, unless ETH fails. © Eduard Eiben, Dušan Knop, Fahad Panolan, and Ondřej Suchý.