  The Cepheid period-luminosity (PL) relation (or Leavitt law) has served as
the first rung of the most widely used extragalactic distance ladder and is
central to the determination of the local value of the Hubble constant ($H_0$).
We investigate the influence of metallicity on Cepheid brightness, a term that
significantly improves the overall fit of the distance ladder, to better define
its wavelength dependence. To this aim, we compare the PL relations obtained
for three Cepheid samples having distinct chemical composition (in the Milky
Way and Magellanic Clouds) and focusing on the use of improved and recent data
while covering a metallicity range of about 1 dex. We estimate the metallicity
effect (hereafter $\gamma$) in 15 filters from mid-IR to optical wavelengths,
including five Wesenheit indices, and we derive a significant metallicity term
in all filters, in agreement with recent empirical studies and models, in the
sense of metal-rich Cepheids being brighter than metal-poor ones. We describe
the contribution of various systematic effects in the determination of the
$\gamma$ term. We find no evidence of $\gamma$ changing over the wavelength
range $0.5-4.5 \, \rm \mu m$, indicating that the main influence of metallicity
on Cepheids is in their luminosity rather than color. Finally, we identify
factors that sharpen the empirical constraints on the metallicity term over
past studies, including corrections for the depth of the Magellanic Clouds,
better-calibrated Cepheid photometry, improved Milky Way extinction estimates,
and revised and expanded metallicity measurements in the LMC.
