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Context: Lupus is recognised as one of the closest star-forming regions, but the lack of trigonometric parallaxes in the pre-Gaia era hampered many studies on the kinematic properties of this region and led to incomplete censuses of its stellar population. Aims: We use the second data release of the Gaia space mission combined with published ancillary radial velocity data to revise the census of stars and investigate the 6D structure of the Lupus complex. Methods: We performed a new membership analysis of the Lupus association based on astrometric and photometric data over a field of 160 deg2 around the main molecular clouds of the complex and compared the properties of the various subgroups in this region. Results: We identified 137 high-probability members of the Lupus association of young stars, including 47 stars that had never been reported as members before. Many of the historically known stars associated with the Lupus region identified in previous studies are more likely to be field stars or members of the adjacent Scorpius-Centaurus association. Our new sample of members covers the magnitude and mass range from G=8 to G=18 mag and from 0.03 to 2.4Msun, respectively. We compared the kinematic properties of the stars projected towards the molecular clouds Lupus 1 to 6 and showed that these subgroups are located at roughly the same distance (about 160~pc) and move with the same spatial velocity. Our age estimates inferred from stellar models show that the Lupus subgroups are coeval (with median ages ranging from about 1 to 3 Myr). The Lupus association appears to be younger than the population of young stars in the Corona-Australis star-forming region recently investigated by our team using a similar methodology. The initial mass function of the Lupus association inferred from the distribution of spectral types shows little variation compared to other star-forming regions.