No, they had their own pantheon, each city having its own gods. Greek influence arrived with Alexander in the late 4th Century BCE, however the Greek pantheon did not prevail. Of course, the various gods were simply parallels of the same functionary, and recognised by various peoples as such eg the high god was Jupiter (Roman) = Zeus (Greek) = Yahweh (Jewish) = Amon (Egyptian) = Enlil (Mesopotamian) = Ahura Mazda (Persian). No one was particularly fussed about it. When Pompey captured Jerusalem, he automatically went to the temple and sacrificed to Yahweh (Jupiter). Religious distinctions (and wars) are a modern invention of the exclusivist children of Yahweh (Judaism, Christianity and Islam).
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In ancient Greek religion, they certainly never died.
Greek Mythology, also called Classical Mythology is the ancient religion of the ancient Greeks: as reported and reduced by Greek poets of old.
It's passed down by retelling and reinventing the stories of ancient Greek religion.
Ancient Greek religion did not have birth-dates for the goddess Hecate.
The Goddess of wisdom, craft and war in ancient Greek religion and mythology