  We perform a complete spectral analysis of the linearized Shakhov model
involving two relaxation times $\tau_{\rm fast}$ and $\tau_{\rm slow}$. Our
results are based on spectral functions derived from the theory of finite-rank
perturbations, which allows us to infer the existence of a critical wave number
$k_{\rm crit}$ limiting the number of discrete eigenvalues above the essential
spectrum together with the existence of a finite-dimensional slow manifold
defining non-local hydrodynamics. We discuss the merging of hydrodynamic modes
as well as the existence of second sound and the appearance of ghost modes
beneath the essential spectrum in dependence of the Prandtl number.
