  We consider the focusing $5$d Hartree equation, which is $L^2$-supercritical,
with finite energy initial data, and investigate the solutions at the
mass-energy threshold. We establish the existence of special solutions
following the work of Duyckaerts-Roudenko [11] for the $3$d focusing cubic
nonlinear Schr\"odinger equation (NLS). In particular, apart from the ground
state solution $Q$, which is global but non-scattering, there exist special
solutions $Q^+$ and $Q^-$, which in one time direction approach $Q$
exponentially, and in the other time direction $Q^+$ blows up in finite time
and $Q^-$ exists for all time, exhibiting scattering behavior. We then
characterize all radial threshold solutions either as scattering and blow up
solutions in both time directions (similar to the solutions under the
mass-energy threshold, see Arora-Roudenko [3]), or as the special solutions
described above. To obtain the existence and classification result, in this
paper we perform a thorough and meticulous investigation of the spectral
properties of the linearized operator associated to the Hartree equation.
