Most first century history is based on Tacitus, Suetonius, Cassius Dio and Josephus. Cassius Dio says nothing. Tacitus mentions Christ, Suetonius mentions Christians, and Josephus has a passage about the brother of Christ (and another long and corrupt passage which probably did so originally).
You will find, however, that the headbangers are ready with excuses to ignore this data.
Only a fool manufactures a silence and then argues that his manufactured absence of evidence is evidence of absence.
They are not so much excuses but alternative explications.
Josephus
The fact no one cites the TF in Josephus until the 4th century, it breaks the flow of the chapter it is in, and there was a copy of Josephus as late as 1600 which didn't have it makes the paragraph suspect.
In
Against Celsus 1.47 Origen records "this writer" (Josephus) ... "in seeking after the cause of the fall of Jerusalem and the destruction of the temple" ... "says nevertheless" ... "that these disasters happened to the Jews as a punishment f
or the death of James the Just, who was a brother of Jesus (called Christ)"
In
Against Celsus 2.13 Origen states "But at that time there were no armies around Jerusalem, encompassing and enclosing and besieging it; for the siege began in the reign of Nero, and lasted till the government of Vespasian,
whose son Titus destroyed Jerusalem, on account, as Josephus says, of James the Just, the brother of Jesus who was called Christ, but in reality, as the truth makes dear, on account of Jesus Christ the Son of God."
Note that the James-Jesus passage we have in Josephus does NOT connect the death of this James with the destruction of Jerusalem, the Temple, or any other disaster despite Origen stating twice that the passage he is referring to
does.
Finally, the James of Josephus died c64 CE by just stoning while Hegesippus, Clement of Alexandria, Eusebius of Caesarea, and Early Christian tradition all had James the Just dying c. 70 CE by being thrown from a battlement, stoned, and finally clubbed to death by passing laundrymen. In fact, Eusebius of Caesarea in his Church History, Book III, ch. 11 clearly writes "After the martyrdom of James and the conquest of Jerusalem which immediately followed..." but there are seven years and four High Priests between these two events so either we have one of the wonkiest definition of "immediately followed" in the history of the world or these are two different James and the "him called Christ" phrase was added to make the connection.
Tacitus
We now know that our oldest copy of Tacitus was tampered with and he was talking about Chr
estians. The problem here is not only was Chrestus a common slave name and used as a title, but Chr
estians may have been the name of cult that followed Serapis (Osiris) per the Hadrian to Servianus 134 CE letter found in the
Historia Augusta (yes that work has problems but historians are stuck using it)
So this could have been a splinter group of Chr
estians (either Osiris or followers of a Messiah calling himself Chrestus), Tacitus could have been just repeating what Christians were telling him, this could have been in reference to followers of another Christ (which Paul warned of in the 50s CE), or ala Josephus we have more tampering.
Suetonius
No one denies Christianity existing in the 1st century but all we get is "punishments were also inflicted on the Christians, a sect professing a new and mischievous religious belief" Moreover, there is no connection between these punishments (in chapter 16) and the Great Fire (chapter 38) and they are presented as examples of "During his reign many abuses were severely punished and put down, and no fewer new laws were made".
The only headbanger material is with regards to anyone who presents these three as "proof" Jesus existed.