Critical aging properties of the two-dimensional spherical model with the long-range interaction r−3

► The non-equilibrium properties depend on the initial magnetization. ► The magnetization has a logarithmic power-law increase in the short-time regime. ► Logarithmic corrections enter into the two-time scaling behavior. ► Three distinct types of aging are found at both criticality and low temperatures. ► The inclusion of long range interaction r−3 tends to the mean field behavior.