Cepheid Metallicity in the Leavitt Law (C-MetaLL) survey: I. HARPS-N@TNG
  spectroscopy of 47 Classical Cepheid and 1 BL Her variables
  Classical Cepheids (DCEPs) are the most important primary indicators of the
extragalactic distance scale. Establishing the dependence on metallicity of
their period--luminosity and period--Wesenheit ($PLZ$/$PWZ$) relations has deep
consequences on the calibration of secondary distance indicators that lead to
the final estimate of the Hubble constant (H$_0$). We collected high-resolution
spectroscopy for 47 DCEPs plus 1 BL Her variables with HARPS-N@TNG and derived
accurate atmospheric parameters, radial velocities and metal abundances. We
measured spectral lines for 29 species and characterized their chemical
abundances, finding very good agreement with previous results. We re-determined
the ephemerides for the program stars and measured their intensity-averaged
magnitudes in the $V,I,J,H,K_s$ bands.
  We complemented our sample with literature data and used the Gaia Early Data
Release 3 (EDR3) to investigate the $PLZ$/$PWZ$ relations for Galactic DCEPs in
a variety of filter combinations. We find that the solution without any
metallicity term is ruled out at more than the 5 $\sigma$ level. Our best
estimate for the metallicity dependence of the intercept of the $PLK_s$,
$PWJK_s$, $PWVK_s$ and $PWHVI$ relations with three parameters, is
$-0.456\pm$0.099, $-0.465\pm$0.071, $-0.459\pm$0.107 and $-0.366\pm$0.089
mag/dex, respectively. These values are significantly larger than the recent
literature. The present data are still inconclusive to establish whether or not
also the slope of the relevant relationships depends on metallicity. Applying a
correction to the standard zero point offset of the Gaia parallaxes has the
same effect of reducing by $\sim$22\% the size of the metallicity dependence on
the intercept of the PLZ/PWZ relations.
