Technically speaking, the Byzantine Empire was the Eastern Roman Empire, that part of the empire that lived on for nearly a thousand years after the Sack of Rome in 476 AD. By that time, the Western Roman Empire had formally tolerated Christianity for over a century.
But in the preceding centuries, the Roman Empire had been officially pagan, worshipping a pantheon of gods including Jupiter, the father of the gods, and Mars, the god of war. At various times throughout the Empire's history, Christians would be subject to outbreaks of persecution, most notably by the Emperor Nero.
From the start, however, the Eastern half of the Roman Empire—or Byzantium, as it would come to be known—formally embraced Christianity. The Emperor Constantine, who had himself converted to Christianity, shifted the balance of power in the Roman Empire towards the East by moving the imperial capital to the city of Byzantium, which he renamed Constantinople after himself.
Under Constantine, Constantinople became one of the great centers of Christianity. That status was consolidated nearly two hundred years after the Emperor's death by the construction of the Hagia Sophia, a magnificent basilica that still stands today, although it is now a mosque. What was once Constantinople is today the city of Istanbul in Turkey.