Sirius has the best-documented practical role of any star in this collection, and it is Egyptian rather than Greek.
The heliacal rising of a star is its first visible appearance in the dawn twilight after a period of invisibility. For Sirius, Sothis in Greek, Sopdet in Egyptian, this occurred in midsummer, and it coincided closely with the annual inundation of the Nile, on which Egyptian agriculture entirely depended. The Egyptians used the event as a calendar marker, and the identification of Sopdet with Isis is old and central.
This produced a real technical problem that is well documented. The Egyptian civil year was 365 days with no leap day, so it drifted against the solar year by about a day every four years, and the civil calendar and the heliacal rising of Sothis came back into agreement only after roughly 1,460 years, the Sothic cycle. Recorded observations of the Sothic rising against the civil calendar are among the anchors used to date Egyptian chronology, though how securely is debated.
The Greek and Roman inheritance is the dog days: the hottest part of summer, coinciding with Sirius's heliacal rising, and blamed on the star. Ancient writers attributed fever, lassitude and madness in dogs to its influence, and the Latin dies caniculares is the source of the English phrase. This is the origin of the Mars component in the star's nature.
Ptolemy assigns Sirius the nature of Jupiter and Mars: Jupiter for honour, wealth and high office, Mars for the heat. Agrippa pairs it with beryl and juniper in the Behenian list.