  In the attempt to fully profile a $Z'$ boson accessible at the Large Hadron
Collider (LHC), we study the sensitivity of di-lepton (for the electron, muon
and tauon cases) and di-quark (for the case of the heavy flavours, $t$ and,
possibly, $b$) samples to the nature of the new gauge state, for a
one-dimensional class of non-exotic $Z'$ bosons. Assuming realistic final state
reconstruction efficiencies and error estimates, we find that, depending on the
CERN collider energy and luminosity, the best chances of extracting the $Z'$
quantum numbers occur when two or more of these channels are simultaneously
explored, as none of them separately enables one to fully probe the parameter
spaces of the aforementioned models. Effects of Standard Model (SM) background
as well interferences between this and the various $Z'$ signals have been
accounted for. A complete study of cross sections and asymmetries (both spatial
and spin ones) makes clear the need for complementarity, especially for their
disentanglement over the full parameter space.
