  We perform a detailed study of the cosmological dynamics of a recently
proposed infrared modification of the Einstein equations, based on the
introduction of a non-local term constructed with $m^2g_{\mu\nu}\Box^{-1} R$,
where $m$ is a mass parameter. The theory generates automatically a dynamical
dark energy component, that can reproduce the observed value of the dark energy
density without introducing a cosmological constant. Fixing $m$ so to reproduce
the observed value $\Omega_{\rm DE}\simeq 0.68$, and writing $w(a)=w_0+(1-a)
w_a$, the model provides a neat prediction for the equation of state parameters
of dark energy, $w_0\simeq -1.042$ and $w_a\simeq -0.020$. We show that,
because of some freedom in the definition of $\Box^{-1}$, one can extend the
construction so to define a more general family of non-local models. However,
in a first approximation this turns out to be equivalent to adding an explicit
cosmological constant term on top of the dynamical dark energy component. This
leads to an extended model with two parameters, $\Omega_{\Lambda}$ and $m$.
Even in this extension the EOS parameter $w_0$ is always on the phantom side,
in the range $-1.33 < w_0\leq -1$, and there is a prediction for the relation
between $w_0$ and $w_a$.
