The statistics of two-dimensional decaying turbulence are investigated by the use of direct numerical simulation in a periodic box. A number of cases are systematically compared in which the incompressible Navier-Stokes equation is solved on various resolutions and for various Reynolds numbers starting with random velocity fields of a prescribed energy spectrum. It is observed that the statistics are contaminated rapidly by the periodicity of the flow field which blocks the transfer of velocity fluctuations towards larger scales. At large Reynolds numbers the enstrophy spectrum decays similarly in time throughout the whole wavenumber range and the enstrophy decreases inversely proportionally to time.