We present a new extended and detailed set of models for Classical Cepheid pulsators at solar chemical composition ($Z=0.02$, $Y=0.28$) based on a well tested nonlinear hydrodynamical approach. In order to model the possible dependence on crucial assumptions such as the Mass-Luminosity relation of central Helium burning intermediate-mass stars or the efficiency of superadiabatic convection, the model set was computed by varying not only the pulsation mode and the stellar mass but also the Mass-Luminosity relation and the mixing length parameter that is used to close the system of nonlinear hydrodynamical and convective equations. The dependence of the predicted boundaries of the instability strip as well as of both light and radial velocity curves on the assumed Mass-Luminosity and the efficiency of superadiabatic convection is discussed. Nonlinear Period-Mass-Luminosity-Temperature, Period-Radius and Period-Mass-Radius relations are also computed. The theoretical atlas of bolometric light curves for both the fundamental and first overtone mode has been converted in the Gaia filters $G$, $G_{BP}$ and $G_{BR}$ and the corresponding mean magnitudes have been derived. Finally the first theoretical Period-Luminosity-Color and Period-Wesenheit relations in the Gaia filters are provided and the resulting theoretical parallaxes are compared with Gaia Data Release 2 results for both fundamental and first overtone Galactic Cepheids.
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