We study the pressure spectrum P(t) of the maximal measure for arbitrary rational maps. We also consider its modified version which is defined by means of the variational principle with respect to non-atomic invariant measures. It is shown that for negative values of t, the modified spectrum has all major features of the hyperbolic case (analyticity, the existence of a spectral gap for the corresponding transfer operator, rigidity properties, etc). The spectrum P(t) can be computed in terms of . Their Legendre transforms are the Hausdorff and the box-counting dimension spectra of the maximal measure respectively. This work is closely related to a paper [32] by D. Ruelle.