For those interested in the truth of the matter, there was sporadic persecution across the Empire - most commonly by regional governors. The essential conflict between Christians and the Roman government was that the Roman government regarded themselves as having a ‘social contract’, if you will, with the gods. “Respect us, and we will bless you” sort of thing. Do ut des. Furthermore, oaths were often sworn by invoking the relevant gods - which makes refusing to swear oaths of allegiance very suspect. Christians were not fond of being forced into this position, understandably.

Some pagan Emperors were positively inclined towards Christianity - Antoninus Pius put Christians under the direct protection of the Emperor, Marcus Aurelius is noted to have had a tolerant view of Christians and contemporary Christians regarded him highly, Philip the Arab participated in Christian religious celebrations. Julian the Apostate viewed Christianity unfavorably, but declared religious tolerance in the Empire (universal tolerance would die with his Christian successors).

Most pagan Emperors were indifferent towards Christianity - for Vespasian, Christians amongst the Iudeans during the First Jewish-Roman War were treated exactly the same as Jews, suggesting that Romans did not see or did not care about any differences with Judaism; Trajan writes (in response to a governor’s letter) that Christians were not to be hunted down or sought out, but if revealed in some other investigation, it was permissible to put non-citizens to death for refusing to recant, not for their faith, but for ‘obstinacy’ towards the proper authorities. Filthy provincials! He is unlikely to have given the issue much more thought than that single letter, considering Christianity was effectively unknown to Romans at the time. Septimius Severus is suggested to have been hostile towards evangelism, but tolerant enough towards those who were already Christian to trust them in high positions of power.

Some pagan Emperors were outright hostile - Nero, infamously, blamed the then-unknown cult of ‘Chrestians’ for the Great Fire of Rome and DID actually feed quite a few to the lions. There seems to be no other incident in which Nero’s reign took note of the new religion - an opportunistic scapegoat. His successor (or usurper, if one prefers), Vespasian, is not recorded to have continued this trend in any way. Domitian and Hadrian are sometimes suggested as persecutors of Christians, but the Romans did not always fully understand the difference between Jews and Christians at this point, and because of that, there is some uncertainty as to whether they persecuted Christians specifically, or simply regarded them as Jews. Diocletian is the worst of the lot, but also the last - he actually did mandate persecutions across the Empire, under which several thousand Christians were killed. His successor, Constantine, was a Christian, and would promptly reverse which faith the Empire favored.