According to § 36 Hessisches Hochschulgesetz (HHG) and § 3 Grundordnung der Philipps-Universität (GrundO) the senate functions as an advisory board in matters of research, teaching and studies, which are pivotal to the university. The senate has the right of initiative and overlooks the management of the chair.
A full list of functions and the current composition can be found here.
In general the senate comprises 17 members. Nine of whom are elected by the ~370 professors, three by the ~24300 students, three by the ~1800 academic staff and another two by the administrative-technical staff. The latest numbers can be found here.
In conclusion, the highest body of the university is dominated by a majority of professors.