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[ABRIDGED] We present the first statistical analysis of the completeESO-Sculptor redshift survey (ESS). The flux-calibrated sample of 617 galaxyspectra with R_c<20.5 is separated into 3 spectral classes (early,intermediate, and late). We report in detail on the spectral classification,the polynomial K-corrections, and all sources of corresponding random andsystematic errors. The derived luminosity functions (LF) in the Johnson-CousinsB and Rc bands are in agreement with the results from the comparable CNOC2survey (Lin et al. 1999), whereas the ESS provides the first estimates of LFsper spectral type in the Johnson V band. A renewed interpretation of the galaxyLFs from a redshift survey are obtained by fitting the ESS LFs with compositefunctions based on the local LFs per morphological type (Sandage, Binggeli &Tammann 1985; Jerjen & Tammann 1997). As good or better fits than with pureSchechter functions are obtained using: for the early-type LF, a two-wingGaussian; for the intermediate-type LF, the sum of a Gaussian modeling theSpiral galaxies and a steep Schechter function (alpha=-1.5) representing thedwarf elliptical galaxies; for the late-type LF, a similar composite functionwith a flat or weaker slope (-0.8<alpha<-0.3) for the Schechter component whichrepresents the dwarf Irregular galaxies. This analysis illustrates how LFs perspectral type may be affected by morphological type mixing, and emphasizes theneed for a quantitative morphological classification at z>0.1 which separatesthe giant and dwarf galaxy populations.Comment: 33 Pages, 13 color figures, A&A Latex macro. Published in A&

The ESO-Sculptor Survey: Luminosity functions of galaxies per spectral type at redshifts $0.1{-}0.5$