The static and high frequency dielectric tensors, Born effective charges, vibrational spectrum at the Γ point, TO-LO splitting and IR intensities of calcite CaCO3 have been calculated with the periodic ab initio CRYSTAL program, with five different basis sets of increasing size and four different Hamiltonians (HF, LDA, PBE, B3LYP). B3LYP is shown to perform better than the other options, in particular of LDA and PBE that are often used for the calculation of the vibrational spectrum of crystalline solids. When comparing B3LYP and experimental frequencies, the mean absolute difference is as small as 8.5 cm-1; this number reduces to 4.8cm-1 if the two lowest experimental frequencies, that we suspect to be affected by a relatively large error, are excluded from statistics. Static and high frequency dielectric tensors, as well as IR intensities computed with the same hybrid scheme (B3LYP) compare quite favourably with experiment. The full set of modes is characterized by various tools including isotopic substitution, "freezing" one of the two subunits (Ca2+ or CO32-) and graphical representations. A general tool has been implemented, that permits the automatic generation of the animation of the full set of modes starting from the CRYSTAL output (available at www.crystal.unito.it/vibs/calcite).