Ohio Rev. Code § 3109.042: Custody rights of unmarried mother.

(A) An unmarried female who gives birth to a child is the sole residential parent and legal custodian of the child until a court of competent jurisdiction issues an order designating another person as the residential parent and legal custodian. A court designating the residential parent and legal custodian of a child described in this section shall treat the mother and father as standing upon an equality when making the designation.

(B) Notwithstanding division (A) of this section, an unmarried female who has been convicted of or pleaded guilty to rape or sexual battery and has been declared under section 3109.501 of the Revised Code to be the parent of a child born as a result of rape or sexual battery shall not be a residential parent and legal custodian of that child.