You can certainly file for it, but that doesn't mean you'll get it. A judge will determine what is in the best interests of the child. If you can agree on a parenting plan, then the court proceedings will go more smoothly. If you can't agree, the judge will decide if shared custody or custody with visitation is best.
Since you weren't married, you may have to establish paternity if you didn't sign the AOP paperwork at the hospital when your son was born.
