  Let $G$ be a (non compact) connected simply connected locally compact second
countable Lie group, either abelian or unimodular of type I, and $\rho$ an
irreducible unitary representation of $G$. Then, we define the analytic torsion
of $G$ localised at the representation $\rho$. Next, let $\Gamma$ a discrete
cocompact subgroup of $G$. We use the localised analytic torsion to define the
relative analytic torsion of the pair $(G,\Gamma)$, and we prove that it
coincides with the Lott $L^2$ analytic torsion of a covering space. We
illustrate these constructions analysing in some details two examples: the
abelian case, and the case $G=H$, the Heisenberg group.
