Classical Cepheids (DCEPs) are the most important primary indicators for the extragalactic distance scale. Establishing the dependence on metallicity of their period--luminosity and period--Wesenheit (PL/PW) relations has deep consequences on the estimate of the Hubble constant (H$_0$). We aim at investigating the dependence on metal abundance ([Fe/H]) of the PL/PW relations for Galactic DCEPs. We combined proprietary and literature photometric and spectroscopic data, gathering a total sample of 413 Galactic DCEPs (372 fundamental mode -- DCEP_F and 41 first overtone -- DCEP_1O) and constructed new metallicity-dependent PL/PW relations in the near infra-red (NIR) adopting the Astrometric Based Luminosity. We find indications that the slopes of the PL$(K_S)$ and PW$(J,K_S)$ relations for Galactic DCEPs might depend on metallicity when compared to the Large Magellanic Cloud relationships. Therefore, we have used a generalized form of the PL/PW relations to simultaneously take into account the metallicity dependence of the slope and intercept of these relations. We calculated PL/PW relations which, for the first time, explicitly include a metallicity dependence of both the slope and intercept terms. Although the insufficient quality of the available data makes our results not yet conclusive, they are relevant from a methodological point of view. The new relations are linked to the geometric measurement of the distance to the Large Magellanic Cloud and allowed us to estimate a {it Gaia} DR2 parallax zero point offset $Delta varpi$=0.0615$pm$0.004 mas from the dataset of DCEPs used in this work.
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