Context. Long gamma-ray bursts (LGRBs) are related to the final stages of evolution of very massive stars. As such, they should follow the star formation rate (SFR) of galaxies. We can use them to probe for star-forming galaxies in the distant universe following this assumption. The relation between the rate of LGRBs in a given galaxy and its SFR (which we call the LGRB bias) may however be complex, as we have good indications that the LGRB hosts are not perfect analogues to the general population of star-forming galaxies. Aims: In this work, we try to quantify how the LGRB bias depends on physical parameters of their host galaxy, such as SFR or stellar mass. These trends may reveal more fundamental properties such as the role of the metallicity of LGRBs and of their progenitors . Methods: We propose an empirical method based on the comparison of stellar mass functions (and SFR distributions) of LGRB hosts and of star-forming galaxies to find how the bias depends on the stellar mass or the SFR. Results: By applying this method to a sample of LGRB hosts at redshifts lower than 1.1, where the properties of star-forming galaxies are fairly well established and where the properties of LGRB host galaxies can be deduced from observations (limiting ourselves to stellar masses higher than 109.25 M⊙ and SFR higher than ~1.8 M⊙ yr-1), we find that the LGRB bias depends on both the stellar mass and SFR. We find that the bias decreases with the SFR; that is, we see no preference for highly star-forming galaxies, once we account for the higher number of massive stars in galaxies with larger SFR. We do not find any trend with the specific star formation rate (SSFR), but the dynamical range in SSFR in our study is narrow. Through an indirect method, we relate these trends to a possible decrease in the LGRBs rate / SFR ratio with the metallicity. Conclusions: The method we propose suggests trends that may be useful to constrain models of LGRB progenitors, showing a clear decrease in the LGRB bias with the metallicity. This is promising for the future as the number of LGRB hosts studied will increase.
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