Context. 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 and PW) relations has deep consequences for the estimate of the Hubble constant ( H0). Aims. We investigate the dependence on metal abundance ([Fe/H]) of the PL and PW relations for Galactic DCEPs. Methods. 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 and PW relations in the near-infrared adopting the astrometry-based luminosity. Results. We find indications that the slopes of the PL( KS) and PW( J, KS) relations for Galactic DCEPs might depend on metallicity on the basis of the comparison with the Large Magellanic Cloud relationships. Therefore we used a generalized form of the PL and PW relations to simultaneously take the metallicity dependence of the slope and intercept of these relations into account. Conclusions. We calculated PL and PW relations that for the first time explicitly include a metallicity dependence of the slope and intercept terms. The quality of the available data is insufficient, however, and we cannot yet present conclusive results, but 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 Gaia DR2 parallax zero-point offset Δ ϖ = 0.0615 ± 0.004 mas from the dataset of DCEPs used in this work.
