Hi every one got a question needing help with

Im tring to size my Grounding electrode conductor. I have a 146amp calculated load and im going to use #1 copper for this. The GEC cant be under two AWG sizes smaller than your main feeder lines right? so if I use a #3 will this be fine? by the way this is a 120/240v 3wire single phase service.

Any help would be great thanks