Context : Zinc in stars is an important reference element because it is a proxy to Fe in studies of damped Lyman- \alpha systems , permitting a comparison of chemical evolution histories of bulge stellar populations and DLAs . In terms of nucleosynthesis , it behaves as an alpha element because it is enhanced in metal-poor stars . Abundance studies in different stellar populations can give hints to the Zn production in different sites . Aims : The aim of this work is to derive the iron-peak element Zn abundances in 56 bulge giants from high resolution spectra . These results are compared with data from other bulge samples , as well as from disk and halo stars , and damped Lyman- \alpha systems , in order to better understand the chemical evolution in these environments . Methods : High-resolution spectra were obtained using FLAMES+UVES on the Very Large Telescope . We computed the Zn abundances using the Zn I lines at 4810.53 and 6362.34 Å . We considered the strong depression in the continuum of the Zn I 6362.34 Å line , which is caused by the wings of the Ca I 6361.79 Å line suffering from autoionization . CN lines blending the Zn I 6362.34 Å line are also included in the calculations . Results : We find [ Zn/Fe ] =+0.24 \pm 0.02 in the range - 1.3 < [ Fe/H ] < - 0.5 and [ Zn/Fe ] =+0.06 \pm 0.02 in the range - 0.5 < [ Fe/H ] < - 0.1 , whereas for [ Fe/H ] \geq - 0.1 , it shows a spread of - 0.60 < [ Zn/Fe ] < +0.15 , with most of these stars having low [ Zn/Fe ] < 0.0 . These low zinc abundances at the high metallicity end of the bulge define a decreasing trend in [ Zn/Fe ] with increasing metallicities . A comparison with Zn abundances in DLA systems is presented , where a dust-depletion correction was applied for both Zn and Fe . When we take these corrections into account , the [ Zn/Fe ] vs. [ Fe/H ] of the DLAs fall in the same region as the thick disk and bulge stars . Finally , we present a chemical evolution model of Zn enrichment in massive spheroids , representing a typical classical bulge evolution . Conclusions :