We have built a new 3D Global Climate Model (GCM) to simulate Pluto as observed by New Horizons in 2015. All key processes are parametrized on the basis of theoretical equations, including atmospheric dynamics and transport, turbulence, radiative transfer, molecular conduction, as well as phases changes for N2, CH2 and CO. Pluto's climate and ice cycles are found to be very sensitive to model parameters and initial states. Nevertheless, a reference simulation is designed by running a fast, reduced version of the GCM with simplified atmospheric transport for 40,000 Earth years to initialize the surface ice distribution and sub-surface temperatures, from which a 28-Earth-year full GCM simulation is performed. Assuming a topographic depression in a Sputnik-planum (SP)-like crater on the anti-Charon hemisphere, a realistic Pluto is obtained, with most N2 and CO ices accumulated in the crater, methane frost covering both hemispheres except for the equatorial regions, and a surface pressure near 1.1 Pa in 2015 with an increase between 1988 and 2015, as reported from stellar occultations. Temperature profiles are in qualitative agreement with the observations. In particular, a cold atmospheric layer is obtained in the lowest kilometers above Sputnik Planum, as observed by New Horizons's REX experiment. It is shown to result from the combined effect of the topographic depression and N2 daytime sublimation. In the reference simulation with surface N2 ice exclusively present in Sputnik Planum, the global circulation is only forced by radiative heating gradients and remains relatively weak. Surface winds are locally induced by topography slopes and by N2 condensation and sublimation around Sputnik Planum. However, the circulation can be more intense depending on the exact distribution of surface N2 frost. This is illustrated in an alternative simulation with N2 condensing in the South Polar regions and N2 frost covering latitudes between 35°N and 48°N. A global condensation flow is then created, inducing strong surface winds everywhere, a prograde jet in the southern high latitudes, and an equatorial superrotation likely forced by barotropic instabilities in the southern jet. Using realistic parameters, the GCM predict atmospheric concentrations of CO and CH4 in good agreement with the observations. N2 and CO do not condense in the atmosphere, but CH4 ice clouds can form during daytime at low altitude near the regions covered by N2 ice (assuming that nucleation is efficient enough). This global climate model can be used to study many aspects of the Pluto environment. For instance, organic hazes are included in the GCM and analysed in a companion paper (Bertrand and Forget, Icarus, this issue).