The Avesta is not written in one language. The Gathas, together with the short Yasna Haptanghaiti, are in Old Avestan, an archaic Iranian language; the rest is in Young Avestan, later and grammatically simpler. The gap between them is large enough that Young Avestan composers appear at points not to have fully understood the older text they were transmitting.
Old Avestan is strikingly close to the language of the Rig Veda, close enough that whole lines can be converted into Vedic Sanskrit by regular sound rules. That is the strongest evidence for dating, and it points early. On the linguistic argument, most Iranists now place the Gathas somewhere between about 1500 and 1000 BCE, which would put Zarathustra in the Bronze Age, before the Iranian tribes reached western Iran and long before any Persian empire.
The traditional chronology says otherwise. Zoroastrian and later Greek and Arabic sources place Zarathustra "258 years before Alexander", yielding a date around 600 BCE and making him roughly contemporary with the rise of the Medes; a separate strand of Greek report placed him thousands of years earlier still, which nobody now defends. The 258-year figure was long accepted and is defended by scholars who read it as preserving a genuine memory rather than a calculation invented to fit later needs, and it remains the date many Zoroastrians themselves give. The dating is therefore genuinely contested, but the weight of it runs early: the linguistic evidence is hard to explain if the Gathas were composed within a century or two of the Achaemenids, and the arguments for the late date rest on transmitted chronology rather than on the text.