Period-luminosity and period-luminosity-metallicity relation for $\delta$ Scuti Stars

$\delta$ Scuti ($\delta$ Sct) stars are potential distance tracers for studying the Milky Way structure. We conduct a comprehensive analysis of the period-luminosity (PL) and period-luminosity-metallicity (PLZ) relation for $\delta$ Sct stars, integrating data from the Zwicky Transient Facility (ZTF), the Transiting Exoplanet Survey Satellite (TESS), Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST), Apache Point Observatory Galactic Evolution Experiment (APOGEE), and Gaia. To mitigate the impact of the Gaia parallax zero point offset, we applied a correction method, determining the optimal zero point value to be $zp_\varpi = 35 \pm 2 \, \mu\text{as}$. Using the three best bands, by varying the parallax error threshold, we found that the total error of the PLR zero point was minimized to 0.9\% at a parallax error threshold of 6\%. With this threshold, we derived the PL and PLZ relation for nine bands (from optical to mid-infrared) and five Wesenheit bands. Through our analysis, we conclude that the influence of metallicity on the PLR of $\delta$ Sct stars is not significant, and the differences across various bands are minimal.