Absolute Properties of the Eclipsing $\gamma$ Dor Star V404 Lyrae

We present the first high-resolution spectra for the eclipsing binary V404 Lyr showing $\gamma$ Dor pulsations, which we use to study its absolute properties. By fitting models to the disentangling spectrum of the primary star, we found that it has an effective temperature of $T_{\rm eff,1}=7,330 \pm 150$ K and a rotational velocity of $v_1\sin$$i=148\pm18$ km s$^{-1}$. The simultaneous analysis of our double-lined radial velocities and the pulsation-subtracted ${\it Kepler}$ data gives us accurate stellar and system parameters of V404 Lyr. The masses, radii, and luminosities are $M_1$ = 2.17$\pm$0.06 M$_\odot$, $R_1$ = 1.91$\pm$0.02 R$_\odot$, and $L_1$ = 9.4$\pm$0.8 L$_\odot$ for the primary, and $M_2$ = 1.42$\pm$0.04 M$_\odot$, $R_2$ = 1.79$\pm$0.02 R$_\odot$, and $L_2$ = 2.9$\pm$0.2 L$_\odot$ for the secondary. The tertiary component orbiting the eclipsing pair has a mass of $M_{\rm 3b}$ = 0.71$\pm$0.15 $M_\odot$ in an orbit of $P_{\rm 3b}$ = 642$\pm$3 days, $e_{\rm 3b}$ = 0.21$\pm$0.04, and $a_{\rm 3b}$ = 509$\pm$2 R$_\odot$. The third light of $l_3=4.1\pm0.2\%$ could be partly attributable to the K-type circumbinary object. By applying a multiple frequency analysis to the eclipse-subtracted light residuals, we detected 45 frequencies with signal to noise amplitude ratios larger than 4.0. Identified as independent pulsation modes, seven frequencies ($f_1-f_6$, $f_9$), their new pulsation constants, and the location in the Hertzsprung-Russell diagram indicate that the pulsating primary is a $\gamma$ Dor-type variable star.