Classical double-mode pulsators (RR Lyrae stars and delta Cepheids) are important for their simultaneous pulsation in low-order radial modes. This enables us to put stringent constraints on their physical parameters. We use 30 bright galactic double-mode RR~Lyrae (RRd) stars to estimate their luminosities and compare them with those derived from the parallaxes of the recent data release (EDR3) of the Gaia survey. We employ pulsation and evolutionary models, together with observationally determined effective temperatures to derive the basic stellar parameters. Excluding 6 outlying stars (e.g., with blending issues) the RRd and Gaia luminosities correlate well. With the adopted temperature zero point from one of the works based on the infrared flux method, we find it necessary to increase the Gaia parallaxes by 0.02 mas to bring the RRd and Gaia luminosities into agreement. This value is consonant with those derived from studies on binary stars in the context of Gaia. We examine also the resulting period-luminosity-metallicity (PLZ) relation in the 2MASS K band as follows from the RRd parameters. This leads to the verification of two independently derived other PLZs. No significant zero point differences are found. Furthermore, the predicted K absolute magnitudes agree within sigma=0.005-0.01mag.
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