He likely never existed , but if he did , I think that its most likely that his Mother was a very beautiful woman that Caesar saw and wanted to have his way with and he did and he impregnated her and somehow found out that she had birthed his child .
Though he was not happy to have a 1/2 jewish son , he couldnt bring himself to have him put to death , so he came up with the idea that if the jews were to recieve a king , a messiah , that it should be of Roman blood .
So , this explains why theres no stories of jesus' childhood , he was away , being taught how to be a messiah and his father , the Caesar , planned out the whole affair of jesus' rise to fame , and even told him that he would not really be sacrificed on the cross, that he would be saved , but something happened , maybe there were too many ppl there to take him away and pretend he died , or maybe Caesar thought it best if he actually did die ?
But this is maybe why jesus felt that his father had forsaken him in his last moments ?

I recall a story about that. Romans kept records. There was one that jesus's dad was a Roman archer that was killed in germany