This paper is concerned with the pinning consensus problem of multi-agent systems with single integrator agents in general directed networks. In this problem, the consensus speed is critical for the analysis of the convergence rate of the system. We show that the upper bound of the consensus speed of static pinning control is the eigenvalue with the smallest real part of the matrix reduced by deleting rows and columns corresponding to indexes of pinned agents from the Laplacian matrix. We also investigate the limit of eigenvalues of the controlled multi-agent system as the control gains go to infinity. In examples, we discuss the two cases where the network topologies are a directed small-scale and a directed scale-free graph.