What was Augustus (the first Roman emperor) function as a god?
During his life, he held the title of ‘Pontifex Maximus’, making him the chief priest. This did not necessarily make him a god, though he was deified posthumously. (There was nothing inherently ‘radical’ about deification of rulers, being established in the 4th century BC and into the Hellenic period, with Alexander the Great and the Diadochoi.) I believe as a god Augustus was ‘looked to’ as a highly important figure, not necessarily being a patron of any specific facet of Roman life (whereas Roman gods, e.g. Mars, the god of war, presided over a specific ‘area’). People would make offerings and prayers to him in the hope of getting something in return; what the worshipper wanted would depend on the individual. However, that Augustus was deified was highly important, because it set the precedent for later emperors also to be made gods after death.