
Christians and their martyrdom
[Candida] Moss, professor of New Testament and early Christianity at the University of Notre Dame, challenges some of the most hallowed legends of the religion when she questions what she calls “the Sunday school narrative of a church of martyrs, of Christians huddled in catacombs out of fear, meeting in secret to avoid arrest and mercilessly thrown to lions merely for their religious beliefs.” None of that, she maintains, is true.

The early Christians were not prosecuted for being Christian. The Roman empire was insanely huge, there were thousands of religions everywhere, so the Romans couldn't care less who did what to or with whom, or why, as long as they stayed within the law and paid their taxes. (Most modern governments can learn a thing or two of that, but that's my personal view.)
What then caused the Christians to be picked off the street and occasionally be thrown for the animals? (Not always lions, often these were wild dogs or bears.) The Christians didn't want to obey many laws of the Romans. They felt they stood higher than the laws of those savages with their many gods. The early Christians were known to insult others, did not want to pay their taxes and other things that Roman law ordered, and that apparently was, at first, most of the issue there.
The PBS site has a long and interesting article on this too. Part of the article (C.E. = Common Era, or "A.D."):
Before the year 250 [C.E.], the persecution of Christians is sporadic. It's local. It's improvised. It is at the discretion of a Governor to whom complaints are made and so on. It's not a dragnet and it's not an imperial policy. After 250, when the empire is being battered on every frontier by invading armies, when there's absolute rampant inflation, [there is] incredible governmental instability. There are an average of two or three Emperors in a year. They keep getting assassinated. It's just an incredibly fraught time. That's also the point at which you begin to get the imperial expression of persecution of Christians. Now then again, also, it's interesting. It's not a criminal offense to be a Christian. What you have to do is get a ticket, a lebevos, a chit saying that you have sacrificed for the well-being of the empire...
The article I couldn't find any more told that the first written proof of Christians being thrown for the lions because of their religion date from somewhere in the 13th or 14th century. Only a good millennium later. Not much for eye witness reporting...
Sorry. I don't believe that Christians were prosecuted for their faith. They were prosecuted here and there for being stuck-up, for wanting to convert others. As far as I know they were among the very few religions that actively sought to convert people to their ideas. Not much has changed there...