The physical origin of the gamma-ray burst (GRB) prompt emission is still a subject of debate. Internal shock models have been widely explored, owing to their ability to explain most of the high-energy properties of this emission phase. While the Band function or other phenomenological functions are commonly used to fit GRB prompt emission spectra, we propose a new parametric function that is inspired by an internal shock physical model. We use this function as a proxy of the model to compare it easily to GRB observations. Methods. We built a parametric function that represents the spectral form of the synthetic bursts provided by our internal shock synchrotron model (ISSM). We simulated the response of the Fermi instruments to the synthetic bursts and fit the obtained count spectra to validate the ISSM function. Then, we applied this function to a sample of 74 bright GRBs detected by the Fermi GBM, and we computed the width of their spectral energy distributions around their peak energy. For comparison, we also fit the phenomenological functions that are commonly used in the literature. Finally, we performed a time-resolved analysis of the broadband spectrum of GRB 090926A, which was jointly detected by the Fermi GBM and LAT. This spectrum has a complex shape and exhibits a power-law component with an exponential cutoff at high energy, which is compatible with inverse Compton emission attenuated by gamma-ray internal absorption. Results. This work proposes a new parametric function for spectral fitting that is based on a physical model. The ISSM function reproduces 81% of the spectra in the GBM bright GRB sample, versus 59% for the Band function, for the same number of parameters. It gives also relatively good fits to the GRB 090926A spectra. The width of the MeV spectral component that is obtained from the fits of the ISSM function is slightly larger than the width from the Band fits, but it is smaller when observed over a wider energy range. Moreover, all of the 74 analyzed spectra are found to be significantly wider than the synthetic synchrotron spectra. We discuss possible solutions to reconcile the observations with the internal shock synchrotron model, such as an improved modeling of the shock microphysics or more accurate spectral measurements at MeV energies.
