Context. We have already imaged a co-moving companion at a projected separation of similar to 260 AU from the young star AB Pic A. Evolutionary model predictions based on JHK photometry of AB Pic b suggest a mass of similar to 13-14 M-Jup, placing the object at the deuterium-burning boundary. Aims. We aim to determine the spectral type, the surface gravity, and the effective temperature of AB Pic b. From the comparison of our absolute photometry to surface fluxes generated by atmospheric models, we also aim at deriving mass and radius estimates that are independent of evolutionary model predictions to test and refine them. Methods. We used the adaptive-optics-fed integral field spectrograph SINFONI to obtain high-quality, medium-resolution spectra of AB Pic b (R-lambda = 1500-2000) over the 1.1-2.5 mu m range. Our analysis relie on comparing our spectra to young standard templates and to the latest libraries of synthetic spectra developed by the Lyon group. Results. AB Pic b is confirmed as a young early-L dwarf companion. We derive a spectral type L0-L1 and find several features indicative of an intermediate gravity atmosphere. A comparison to synthetic spectra yields T-eff = 2000(-300)(+100) K and log(g) = 4 +/- 0.5 dex. Determination of the derived atmospheric parameters of AB Pic b is limited by an imperfect match of current atmosphere spectra with our near-infrared observations of AB Pic b. The current treatment of dust settling and the missing molecular opacity lines in the atmosphere models could be responsible. By combining the observed photometry, the surface fluxes from atmosphere models and the known distance of the system, we derive new mass, luminosity, and radius estimates of AB Pic b. They independently confirm the evolutionary model predictions. We finally review the current methods used for characterizing planetary mass companions and discuss them in the perspective of future planet deep-imaging surveys that will be faced with the same limitations.