LGRBs are associated with massive stars and are therefore linked to star formation. The conditions necessary to produce LGRBs can affect the relation between the LGRB rate and star formation. By using the power of a complete LGRB sample, our aim is to understand whether such a bias exists and, if it does, what is its origin. In this first paper, we build the SED of the z<1 host galaxies of the BAT6 LGRB sample, and determine their stellar masses from SED fitting. We compare the resulting stellar mass distribution (i) with star-forming galaxies observed in deep surveys (UltraVISTA); (ii) with semi-analitical models of the z<1 star forming galaxy population and (iii) with numerical simulations of LGRB hosts having different metallicity thresholds for the progenitor star environment. We find that at z<1 LGRBs tend to avoid massive galaxies and are powerful in selecting faint low-mass star-forming galaxies. The stellar mass distribution of the hosts is not consistent with that of the UltraVISTA star-forming galaxies weighted for their SFR. This implies that, at least at z<1, LGRBs are not unbiased tracers of star formation. To make the two distributions consistent, a much steeper faint-end of the mass function would be required, or a very shallow SFR-Mass relation for the low mass galaxy population. GRB host galaxy simulations indicates that, to reproduce the stellar mass distribution, a metallicity threshold of the order of Z_th=0.3-0.5Z_sun is necessary. Models without a metallicity threshold or with an extreme threshold of Z_th = 0.1Z_sun are excluded at z<1. The use of the BAT6 complete sample makes this result not affected by possible biases which could have influenced past results based on incomplete samples. The preference for low metallicities (Z<~0.5Z_sun) can be a consequence of the particular conditions needed for the progenitor star to produce a GRB. (Abridged)
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