Artemis's study is the study of Artemis Fowl II and formerly the study of Artemis Fowl I. It first appears in Artemis Fowl (book) and is referenced throughout the series. It is the nerve center of the Fowl estate.
Presumably, it is connected to the security room and the safe room, as well as Artemis II's bedroom. It is keeping with the classical appearance of Fowl Manor.
It contains several Apple computers and a projector. In the first book, it is projectng CNN onto a blank wall. It presumably contains the hard drives for Artemis's servers.