Let \(D\) be a finite and simple digraph with vertex set \(V(D)\), and let \(f: V(D) \to \{-1, 1\}\) be a two-valued function. If \(\sum_{x \in N_D^-[v]} f(x) \geq 1\) for each \(v \in V(D)\), where \(N_D^-[v]\) consists of \(v\) and all vertices of \(D\) from which arcs go into \(v\), then \(f\) is a signed dominating function on \(D\). The sum \(\sum_{v \in V(D)} f(v)\) is called the weight of \(f\). The signed domination number, denoted by \(\gamma_S(D)\), of \(D\) is the minimum weight of a signed dominating function on \(D\). In this work, we present different lower bounds on \(\gamma_S(D)\) for general digraphs, show that these bounds are sharp, and give an improvement of a known lower bound obtained by Karami in 2009 [H. Karami, S.M. Sheikholeslami, A. Khodkar, Lower bounds on the signed domination numbers of directed graphs, Discrete Math. 309 (2009), 2567-2570]. Some of our results are extensions of well-known properties of the signed domination number of graphs.