ipod has been restored still will not turn on without being plugged in

Since you have already done a Restore, it sounds like a hardware problem related to the battery not charging (or not holding a charge). If it is less than one year from purchase date, it is under warranty. You should have it checked by an Apple authorized tech. If you have an Apple Store nearby, you can take it to the store's Genius Bar. To make a reservation (to avoid a long wait), you can use the store's web page

http://www.apple.com/retail/

Otherwise, you can use the Contact Us link at the bottom right corner of this web page to contact Apple's tech support.