We perform a comparative ab-initio study of Mn-doped Germanium semiconductor using the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional, DFT+U and Heyd-ScuseriaErnzerhof hybrid functional (HSE). We show that the HSE functional is able to correctly account for the relevant ground state properties of the host matrix as well as of Mn-doped semiconductor. Although the DFT+U and the HSE description are very similar, some differences still remain. In particular, the half-metallicity is lost using DFT+U when a suitable U value, tuned to recover the photoemission spectra, is employed. For comparison, we also discuss the case of Mn in Silicon.