We present results from the analysis of 401 RR Lyrae stars (RRLs) belonging to the field of the Milky Way (MW). For a fraction of them multi-band ($V$, $K_{rm s}$, $W1$) photometry, metal abundances, extinction values and pulsation periods are available in the literature and accurate trigonometric parallaxes measured by the Gaia mission alongside Gaia $G$-band time-series photometry have become available with the Gaia second data release (DR2) on 2018 April 25. Using a Bayesian fitting approach we derive new near-, mid-infrared period-absolute magnitude-metallicity ($PMZ$) relations and new absolute magnitude-metallicity relations in the visual ($M_V - {rm [Fe/H]}$) and $G$ bands ($M_G - {rm [Fe/H]}$), based on the Gaia DR2 parallaxes. We find the dependence of luminosity on metallicity to be higher than usually found in the literature, irrespective of the passband considered. Running the adopted Bayesian model on a simulated dataset we show that the high metallicity dependence is not caused by the method, but likely arises from the actual distribution of the data and the presence of a zero-point offset in the Gaia parallaxes. We infer a zero-point offset of $-0.057$ mas, with the Gaia DR2 parallaxes being systematically smaller. We find the RR Lyrae absolute magnitude in the $V$, $G$, $K_{rm s}$ and $W1$ bands at metallicity of [Fe/H]=$-1.5$ dex and period of P = 0.5238 days, based on Gaia DR2 parallaxes to be $M_V = 0.66pm0.06$ mag, $M_G = 0.63pm0.08$ mag, $M_{K_{rm s}} = -0.37pm0.11$ mag and $M_{W1} = -0.41pm0.11$ mag, respectively.
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