<p>We study non-equilibrium properties of a chain of <i>N</i> oscillators with both long-ranged harmonic interactions and long-range conservative noise that exchange momenta of particle pairs. We derive exact expressions for the (deterministic) energy-current auto-correlation at equilibrium, based on the kinetic approximation of the normal mode dynamics. In all cases the decay is algebraic in the thermodynamic limit. We distinguish four distinct regimes of correlation decay depending on the exponents controlling the range of deterministic and stochastic interactions. Surprisingly, we find that long-range noise breaks down the long-range correlations characteristic of low dimensional models, suggesting a normal regime in which heat transport becomes diffusive. For finite systems, we do also derive exact expressions for the finite-size corrections to the algebraic decay of the correlation. In certain regimes, these corrections are considerably large, rendering hard the estimation of transport properties from numerical data for the finite chains. Our results are tested against numerical simulations, performed with an efficient algorithm.</p>