  Aims: Our objective is the optical and near-infrared spectroscopic
characterisation of 2MASS J0249-0557 c, a recently discovered young planetary
mass companion to the $\beta$ Pictoris member 2MASS J0249-0557. Methods: Using
the Visible and Infrared Survey Telescope for Astronomy (VISTA) Hemisphere
Survey (VHS) and the Two Micron All Sky Survey (2MASS) data, we independently
identified the companion 2MASS J0249-0557 c. We obtained low-resolution optical
spectroscopy of this object using the Optical System for Imaging and
low-intermediate-Resolution Integrated Spectroscopy (OSIRIS) spectrograph at
the Gran Telescopio Canarias (GTC), and near-infrared spectroscopy using the
Son of Isaac (SofI) spectrograph on the New Technology Telescope (NTT).
Results: We classified 2MASS J0249-0557 c with a spectral type of L2.5$\pm$0.5
in the optical and L3$\pm$1 in the near-infrared. We identified spectroscopic
indicators of youth that are compatible with the age of the $\beta$ Pictoris
moving group. We also detect a strong H$\alpha$ emission, with a pEW of
-90$^{+20}_{-40}$A, which seems persistent in time. This indicates strong
chromospheric activity or disk accretion. Although many M-type brown dwarfs
have strong H$\alpha$ emission, this target is one of the very few L-type
planetary mass objects in which this strong H$\alpha$ emission has been
detected. Lithium absorption at 6708 A is observed with pEW $\lesssim$ 5A. We
also computed the binding energy of 2MASS J0249-0557 c and obtained an
(absolute) upper limit of $U=(-8.8\pm4.4) 10^{32}$ J. Conclusions: Similarly to
other young brown dwarfs and isolated planetary mass objects, strong H$\alpha$
emission is also present in young planetary mass companions at ages of some
dozen million years. We also found that 2MASS J0249-0557 c is one of the wide
substellar companions with the lowest binding energy known to date.
