This is a question that must be explored. A Yes to the question would give a huge boost to the reliability of the Gospels.
With this article (Part Five) we turn a corner away from archaeology and non-Christian written references to Gospel persons (the last three articles). Now we discuss the preservation of Jesus’ ministry — his words and activity — after his crucifixion (and resurrection) and up to the time when the Gospels were written.
The answer is of course yes. Now find out the historical evidence. If you have a son or daughter or a co-worker or husband who challenges you, send him or her to this link.
The Synoptics are Matthew, Mark, and Luke. Archaeology affirms their reliability. This post lists some discoveries.