Context. The Sagittarius dwarf spheroidal galaxy (Sgr dSph) is a satellite orbiting the Milky Way that has experienced multiple stripping events due to tidal interactions with our Galaxy. Its accretion history has led to a distinct stellar overdensity, which is the remnant of the core of the progenitor. Aims. We present a complete chemical analysis of 111 giant stars in the core of Sgr dSph to investigate the chemical evolution and enrichment history of this satellite. Methods. Employing the metallicity-sensitive Ca H&K photometry from the Pristine Inner Galaxy Survey, we selected stars that span a wide metallicity range and obtained high-resolution spectra with the ESO FLAMES/GIRAFFE multiobject spectrograph. For the stellar sample covering -2.13 < [Fe/H] < -0.35, we derived abundances for up to 14 chemical elements with average uncertainties of ∼0.09 dex and a set of stellar ages that allowed us to build an age-metallicity relation (AMR) for the entire sample. Results. With the most comprehensive set of chemical species measured for the core of Sgr (Na, Mg, Al, Si, Ca, Sc, Ti, V, Cr, Co, Ba, La, and Eu), we studied several [X/Fe] ratios. Most trends align closely with Galactic chemical trends, but notable differences emerge in the heavy n-capture elements, which offer independent insights into the star formation history of a stellar population. Conclusions. The deficiency in α elements with respect to the Milky Way suggests a slower, less efficient early star formation history, similar to other massive satellites. S -process element patterns indicate significant enrichment from Asymptotic giant branch stars over time. The AMR and chemical ratios point to an extended star formation history, with a rapid early phase in the first gigayears, followed by declining activity and later star-forming episodes. These findings are consistent with Sgr hosting multiple stellar populations, from young (∼4 Gyr) to old, metal-poor stars (∼10 Gyr).