No major religious founder has a less certain date. Estimates for Zarathushtra range from roughly 1500 BCE to the sixth century BCE, and the gap is not a rounding error.
The traditional Zoroastrian date places him 258 years before Alexander, which yields the early sixth century BCE. That figure appears in late sources and is widely regarded as a construction, possibly built backward from the Seleucid era. Against it stands the linguistic argument, which most specialists now find decisive: the Gathas, the seventeen hymns attributed to him personally, are in an archaic Avestan closely comparable in grammar and vocabulary to the language of the Rig Veda. If the Rig Veda belongs to the second millennium BCE, so plausibly does Gathic Avestan. The society the Gathas describe, pastoral, chariot-using, with no cities, no writing, no Persians and no Medes, points the same way.
Mary Boyce argued for a date as early as 1500-1200 BCE. Others place him around 1000 BCE. A minority continue to defend a later date, and Almut Hintze and others have argued the linguistic evidence is less conclusive than it appears because archaic religious language can be preserved artificially.
The place is equally open, somewhere in eastern Iran or Central Asia rather than in the Persian southwest.
What is not in doubt is the Gathas themselves. They are difficult, dense, and in the first person, and they present a man arguing with rival priests, complaining that he has no support and few cattle, and asking Ahura Mazda direct and rather aggrieved questions.