Phone turns off after being unplugged from charger

So the phone was working perfectly but I was out the other day and it hit 1% and died as every other phone. I charged it when I got home but it keep looping i thought you know the battery is dead So I replaced it. After replacing the battery it came on showing charging but turned off after 5seconds. I leaves it for around 3 hours to charge and still just keeps turning off when I unplug the charger. I though it might be the software but I tried to update it in iTunes it wouldnt work I tried restoring and still until I placed it in DFU and updated the firmware it came back to life but still turns off after disconnected and also the battery was saying %4 when i replaced it but after it restarted it shows 100% up to today.