We have performed a systematic study of the ground state properties of the zinc-blende, rock-salt, tetragonal, cuprite, fluorite and pyrite phases of platinum nitride by using the plane wave pseudopotential calculations within the density functional theory. The equilibrium structural parameters and bulk moduli are computed within both the local density approximation (LDA) and generalized gradient approximation (GGA). The comparison of the equation of state (EOS) calculated within the LDA for the pyrite structure with the experimental results demonstrates an excellent agreement, hence the use of the LDA rather than the GGA is essential. Complete sets of elastic moduli are presented for cubic forms. The analysis of the results reveal that the pyrite phase with PtN2 stoichiometry leads to the formation of a hard material with the shear modulus G = 206 GPa. The electronic structure of pyrite PtN2 is given, which shows a narrow indirect gap. The vibrational properties of platinum nitride are investigated in detail from lattice dynamical calculations. The calculations show that fluorite and pyrite structures are dynamically stable as well. However, the calculated vibrational modes of pyrite PtN2 do not show complete agreement with experimental Raman frequencies.