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We present new period-luminosity and period-luminosity-metallicity relations for Galactic RR Lyrae stars based on a sample of 28 pulsators located at distances of up to 1.5 kpc from the Sun. Near-infrared photometry was obtained at the Cerro Armazones Observatory, and parallaxes were taken from the Gaia Early Data Release 3. Relations were determined for the 2MASS JHK_s bands and the W_JK Wesenheit index. We compare our results with other calibrations available in the literature and obtain very good agreement with the photometry of RR Lyraes from the Large Magellanic Cloud (LMC) anchored using the distance to the Cloud, which is based on detached eclipsing binaries. We find that the dependence of absolute magnitudes on metallicity of 0.070 ± 0.042 (J band) to 0.087 ± 0.031 (W_JK index) mag dex^–1 for the population of fundamental pulsators (RRab) is in agreement with previously published phenomenological works. We perform a refined determination of distance to the LMC based on our new calibration and photometry from Szewczyk et al. We study the dependence of the fitted parameters of the fiducial relations and the LMC distance on the systematic parallax offset.