Aims. We extend the relationship between X-ray luminosity (L_x) and rotation period (P_rot) found for main-sequence FGK stars, and test whether it also holds for early M dwarfs, especially in the non-saturated regime (L_x ∝ P_rot⁻²) which corresponds to slow rotators. Methods. We use the luminosity coronal activity indicator (L_x) of a sample of 78 early M dwarfs with masses in the range from 0.3 to 0.75 M_⊙ from the HArps-N red Dwarf Exoplanet Survey (HADES) radial velocity (RV) programme collected from ROSAT and XMM-Newton. The determination of the rotation periods (P_rot) was done by analysing time series of high-resolution spectroscopy of the Ca II H & K and Hα activity indicators. Our sample principally covers the slow rotation regime with rotation periods from 15 to 60 days. Results. Our work extends to the low mass regime the observed trend for more massive stars showing a continuous shift of the L_x∕L_bol versus P_rot power law towards longer rotation period values, and includes a more accurate way to determine the value of the rotation period at which the saturation occurs (P_sat) for M dwarf stars. Conclusions. We conclude that the relations between coronal activity and stellar rotation for FGK stars also hold for early M dwarfs in the non-saturated regime, indicating that the rotation period is sufficient to determine the ratio L_x∕L_bol.