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Your phone was somehow booted into Downloading mode. This happens if you turn it on while holding down Volume down button (volume down and home button). That is a mode that allow you to download software from computer. Even it was showing Downloading, nothing was downloading actually. All you had to do was, hold down Volume Down and Power or Volume Down + Up and Power button at same time and it would've rebooted.
Since the battery was died, it is probably got out of that mode. So after charging, just turn it on and it will work without any issues.
If it is still showing Downloading, then hold down Volume Down and Power until it reboots. If that didn't work, then hold down Volume Up, Down and Power at same time until it goes off and when you see logo, release the buttons and let it load.
Note : If you have any case/cover installed on the phone, remove it and try the steps above.
Please let me know if that didn't work, so we can continue troubleshooting