(Abridged) We have conducted a detailed investigation of the broad-band spectral properties of the gamma-ray selected blazars of the Fermi LAT Bright AGN Sample (LBAS). By combining our accurately estimated Fermi gamma-ray spectra with Swift, radio, infra-red, optical and other hard X-ray/gamma-ray data, collected within three months of the LBAS data taking period, we were able to assemble high-quality and quasi-simultaneous Spectral Energy Distributions (SED) for 48 LBAS blazars.The SED of these gamma-ray sources is similar to that of blazars discovered at other wavelengths, clearly showing, in the usual Log $ u $ - Log $ u$ F$_ u$ representation, the typical broad-band spectral signatures normally attributed to a combination of low-energy synchrotron radiation followed by inverse Compton emission of one or more components. We have used these SEDs to characterize the peak intensity of both the low and the high-energy components. The results have been used to derive empirical relationships that estimate the position of the two peaks from the broad-band colors (i.e. the radio to optical and optical to X-ray spectral slopes) and from the gamma-ray spectral index. Our data show that the synchrotron peak frequency $ u_p^S$ is positioned between 10$^{12.5}$ and 10$^{14.5}$ Hz in broad-lined FSRQs and between $10^{13}$ and $10^{17}$ Hz in featureless BL Lacertae objects.We find that the gamma-ray spectral slope is strongly correlated with the synchrotron peak energy and with the X-ray spectral index, as expected at first order in synchrotron - inverse Compton scenarios. However, simple homogeneous, one-zone, Synchrotron Self Compton (SSC) models cannot explain most of our SEDs, especially in the case of FSRQs and low energy peaked (LBL) BL Lacs. (...)
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