Aims. The atmosphere of Jupiter is characterized by banded jets, including an equatorial super-rotating jet, by an intense moist con-vective activity, and by perturbations exerted by vortices, waves, and turbulence. Even after space exploration missions to Jupiter and detailed numerical modeling of Jupiter, questions remain about the mechanisms underlying the banded jets and the role played by dry and moist convection in maintaining these jets. Methods. We report three-dimensional simulations of the Jupiter weather layer using a global climate model (GCM) called Jupiter-DYNAMICO, which couples hydrodynamical integrations on an icosahedral grid with detailed radiative transfer computations. We added a thermal plume model for Jupiter that emulates the effect of mixing of heat, momentum, and tracers by dry and moist convec-tive plumes that are left unresolved in the GCM mesh spacing with a physics-based approach. Results. Our Jupiter-DYNAMICO global climate simulations show that the large-scale Jovian flow, in particular the jet structure, could be highly sensitive to the water abundance in the troposphere and that an abundance threshold exists at which equatorial super-rotation develops. In contrast to our dry (or weakly moist) simulations, simulations that include the observed amount of tropospheric water exhibit a clear-cut super-rotating eastward jet at the equator and a dozen eastward mid-latitude jets that do not migrate poleward. The magnitudes agree with the observations. The convective activity simulated by our thermal plume model is weaker in the equatorial regions than in mid to high latitudes, as indicated by lightning observations. Regardless of whether they are dry or moist, our simulations exhibit the observed inverse energy cascade from small (eddies) to large scales (jets) in a zonostrophic regime.