iPhone 5S battery percentage incorrect after replacement I've recently replaced the battery on my iPhone 5S (running latest iOS 10.2.1) and the battery percentage has been wrong ever since.

It originally showed around 50% (which I believe is what the original battery was at when I did the replacement), I then discharged the phone (which turned off with the battery still at 50%), then let it charge fully to 100% (when it came back up it showed 50% instead of the expected 1% or so).

I then let it drain again and it shut down at 90% (though battery life was great as it lasted several hours of full brightness + graphics intensive gaming), when it came back up it showed 90% again instead of around 1%. I let it charge fully.

Finally today I once again used the phone (expecting the battery meter to be calibrated) and despite great battery life the lowest the meter ever showed was 90% and it shut down. It's now on charge and came back up at around 90%.

One odd thing I noticed is that when it dies it shows the low battery screen when plugged in but only for a very short time before going straight to the Apple logo, where I would expect it to stick around on the charging screen for a few minutes at least.

I haven't restored it since the battery swap, I am not sure whether I should do that (I'd like to avoid it as it doesn't remember any of the e-mail nor wireless passwords and it's a pain to set them up again).

Regards.
It's hard to know exactly what's going on, but I'd check this app to make sure it reports good health: https://itunes.apple.com/us/app/battery-life-check-devices-runtimes/id1080930585?mt=8.
If everything is good there, then I'd perform a hard reset, which you can do by holding the power and home buttons until the phone restarts and shows the Apple logo.