We present the largest, publicly available, sample of Damped Lyman-$alpha$ systems (DLAs) along Gamma-ray Bursts (GRB) line of sights in order to investigate the environmental properties of long GRBs in the $z=1.8-6$ redshift range. Compared with the most recent quasar DLAs sample (QSO-DLA), our analysis shows that GRB-DLAs probe a more metal enriched environment at $zgtrsim3$, up to $[X/H]sim-0.5$. In the $z=2-3$ redshift range, despite the large number of lower limits, there are hints that the two populations may be more similar (only at 90% significance level). Also at hiz, the GRB-DLA average metallicity seems to decline at a shallower rate than the QSO-DLAs: GRB-DLA hosts may be polluted with metals at least as far as $sim 2$kpc from the GRB explosion site, probably due to previous star-formation episodes and/or supernovae explosions. This shallow metallicity trend, extended now up to $zsim5$, confirms previous results that GRB hosts are star-forming and have, on average, higher metallicity than the general QSO-DLA population. Finally, our metallicity measurements are broadly consistent with the hypothesis of two channels of GRB progenitors, one of which is mildly affected by a metallicity bias. The metallicity evolution of modeled GRB hosts agrees reasonably well with our data up to intermediate redshift, while more data are needed to constrain the models at $zgtrsim 4$.
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