International audienceContext. M dwarfs are an important source of information when studying and probing the lower end of the Hertzsprung-Russell (HR) diagram, down to the hydrogen-burning limit. Being the most numerous and oldest stars in the galaxy, they carry fundamental information on its chemical history. The presence of molecules in their atmospheres, along with various condensed species, complicates our understanding of their physical properties and thus makes the determination of their fundamental stellar parameters more challenging and difficult. Aim. The aim of this study is to perform a detailed spectroscopic analysis of the high-resolution H-band spectra of M dwarfs in order to determine their fundamental stellar parameters and ...