Notice the verses in question and how Jesus explains it. The conviction of sin is because they do not believe in Jesus. What saint of God doesn't believe in Jesus? I find the next part of the statement about conviction of righteousness because Jesus left and judgment because the ruler of the world is judged extremely thought provoking.
7 Nevertheless I tell you the truth. It is to your advantage that I go away; for if I do not go away, the Helper will not come to you; but if I depart, I will send Him to you. 8 And when He has come, He will convict the world of sin, and of righteousness, and of judgment: 9 of sin, because they do not believe in Me; 10 of righteousness, because I go to My Father and you see Me no more; 11 of judgment, because the ruler of this world is judged.