We spectroscopically characterize the atmosphere of HD 106906b, a young low-mass companion near the deuterium burning limit. The wide separation from its host star of 7.1 makes it an ideal candidate for high S/N and high-resolution spectroscopy. We aim to derive new constraints on the spectral type, effective temperature, and luminosity of HD106906b and also to provide a high S/N template spectrum for future characterization of extrasolar planets. We obtained 1.1-2.5 $mu$m integral field spectroscopy with the VLT/SINFONI instrument with a spectral resolution of R~2000-4000. New estimates of the parameters of HD 106906b are derived by analyzing spectral features, comparing the extracted spectra to spectral catalogs of other low-mass objects, and fitting with theoretical isochrones. We identify several spectral absorption lines that are consistent with a low mass for HD 106906b. We derive a new spectral type of L1.5$pm$1.0, one subclass earlier than previous estimates. Through comparison with other young low-mass objects, this translates to a luminosity of log($L/L_odot$)=$-3.65pm0.08$ and an effective temperature of Teff=$1820pm240$ K. Our new mass estimates range between $M=11.9^{+1.7}_{-0.8} M_{rm Jup}$ (hot start) and $M=14.0^{+0.2}_{-0.5} M_{rm Jup}$ (cold start). These limits take into account a possibly finite formation time, i.e., HD 106906b is allowed to be 0--3 Myr younger than its host star. We exclude accretion onto HD 106906b at rates $dot{M}>4.8times10^{-10} M_{rm Jup}$yr$^{-1}$ based on the fact that we observe no hydrogen (Paschen-$beta$, Brackett-$gamma$) emission. This is indicative of little or no circumplanetary gas. With our new observations, HD 106906b is the planetary-mass object with one of the highest S/N spectra yet. We make the spectrum available for future comparison with data from existing and next-generation (e.g., ELT and JWST) spectrographs.
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