The recently observed excess in gamma-ray signal near the Galactic center suggests that dark matter particles may annihilate into charged fermions that produce gamma-ray to be observed. In this paper, we consider a leptonic dark matter, which annihilates into the standard model leptons, μ+μ− and τ+τ−, by the interaction of the gauged lepton number U(1)Lμ−Lτ and fits the observed excess. Interestingly, the necessary annihilation cross section for the observed gamma-ray flux provides a good fit to the value for the relic abundance of dark matter. We identify the preferred parameter space of the model after taking the existing experimental constraints from the precision measurements including the muon (g−2), tau decay, neutrino trident production, dark matter direct detection, LHC, and LEP experiments.