We present a deep study of the average hard X-ray spectra of Seyfert galaxies. We analyzed all public INTEGRAL IBIS/ISGRI data available on all the 165 Seyfert galaxies detected at z<0.2. Our final sample consists of 44 Seyfert 1s, 29 Seyfert 1.5s, 78 Seyfert 2s, and 14 Narrow Line Seyfert 1s. We derived the average hard X-ray spectrum of each subsample in the 17-250keV energy range. All classes of Seyfert galaxies show on average the same nuclear continuum, as foreseen by the zeroth order unified model, with a cut-off energy of Ec>200keV, and a photon index of Gamma ~1.8. Compton-thin Seyfert 2s show a reflection component stronger than Seyfert 1s and Seyfert 1.5s. Most of this reflection is due to mildly obscured (10^23 cm^-2 < NH < 10^24 cm^-2) Seyfert 2s, which have a significantly stronger reflection component (R=2.2^{+4.5}_{-1.1}) than Seyfert 1s (R<=0.4), Seyfert 1.5s (R<= 0.4) and lightly obscured (NH < 10^23 cm^-2) Seyfert 2s (R<=0.5). This cannot be explained easily by the unified model. The absorber/reflector in mildly obscured Seyfert 2s might cover a large fraction of the X-ray source, and have clumps of Compton-thick material. The large reflection found in the spectrum of mildly obscured Seyfert 2s reduces the amount of Compton-thick objects needed to explain the peak of the cosmic X-ray background. Our results are consistent with the fraction of Compton-thick sources being ~10%. The spectra of Seyfert 2s with and without polarized broad lines do not show significant differences, the only difference between the two samples being the higher hard X-ray and bolometric luminosity of Seyfert 2s with polarized broad lines. The average hard X-ray spectrum of Narrow line Seyfert 1s is steeper than those of Seyfert 1s and Seyfert 1.5s, probably due to a lower energy of the cutoff.
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