iphone 4s wont turn on unless plugged in

HHEEEELLLPPP......only bought the phone in Jan 12. Got iphone 4s. I havent used it or had it switched on for few weeks...then today tried to turn it on but it wouldnt...thought battery was low so connected to lap top but it didnt do anything. So i plugged it in to power supply and it comes on and said battery was almost full (86%) it kept turning itself on/off then as soon as unplug it it turns back off.

I managed to get it whilst plugged in to laptop to start restoring it but then it stops with error code 1602.....my iphone is now in recovery mode but still doesnt switch on or anything when plug into laptop until i hold power/home buttons together.....and comes on when plugged into power plug.

Any idea wat problem could be or how to fix it????