Someone who has an FAA ATPL should be able to answer this question, except if he/she had a quicky check.
Marker inbound wrote it clearly; medical is tied to the type of ops; if flying PAYING passengers ( common carriage) on a airplane that needs ATPL for Pic, need First class valid; If being paid for flying ,non-paying pax ( as your boss), or involved in ops (like your M or VP reg) need Commercial privileges therefor 2nd Class medical.
If flying your own BBJ, a third class medical is enough...even on your ATP license