  A one-dimensional long-range model of classical rotators with an extended
degree of complexity, as compared to paradigmatic long-range systems, is
introduced and studied. Working at constant density, in the thermodynamic limit
one can prove the statistical equivalence with the Hamiltonian Mean Field model
(HMF) and $\alpha$-HMF: a second order phase transition is indeed observed at
the critical energy threshold $\varepsilon_c=0.75$. Conversely, when the
thermodynamic limit is performed at infinite density (while keeping the length
of the hosting interval $L$ constant), the critical energy $\varepsilon_c$ is
modulated as a function of $L$. At low energy, a self-organized collective
crystal phase is reported to emerge, which converges to a perfect crystal in
the limit $\epsilon \rightarrow 0$. To analyze the phenomenon, the equilibrium
one particle density function is analytically computed by maximizing the
entropy. The transition and the associated critical energy between the gaseous
and the crystal phase is computed. Molecular dynamics show that the crystal
phase is apparently split into two distinct regimes, depending on the the
energy per particle $\varepsilon$. For small $\varepsilon$, particles are
exactly located on the lattice sites; above an energy threshold
$\varepsilon{*}$, particles can travel from one site to another. However,
$\varepsilon{*}$ does not signal a phase transition but reflects the finite
time of observation: the perfect crystal observed for $\varepsilon >0$
corresponds to a long lasting dynamical transient, whose life time increases
when the $\varepsilon >0$ approaches zero.
