Sony Z2 works only while charging. Battery dead?

[AC Question]

My Z2 turns off automatically.
First a message was shown that the battery is empty.
After a few days this message didn't show, the display just went black and the phone turned off.
This happened mostly then I started an app like Chrome.

When Z2 is connected to a charger, it works fine.
The battery graph shows 100% than instantly 0% and after a full charge 100% again.

I assume that the battery is dead or that I have a loose contact.

I have had my Z2 for almost 2 years. It is my first smartphone, so I don't know if a battery lifetime of 2 years is average or not.
I didn't drop my phone.
It started overnight.
About 6 months ago the display was changed.
Cyanogenmod 12.1 (Android 5.1.1)

What could be wrong?
Should I change the battery?

CPU-Z shows:
Health: Good
Temperature: 25.6 C
Voltage 4290 mV
Seems normal to me.

 

[Rukbat]

I have had my Z2 for almost 2 years. It is my first smartphone, so I don't know if a battery lifetime of 2 years is average or not.

Anywhere from 4 months to a decade or two is "average - it depends on how the battery was treated. (Use the last ounce of power every charge cycle and it'll last a few months. Never let it go below 40% and ... well,my 11 year old battery is still working fine, so I don't know the maximum lifetime.)

About 6 months ago the display was changed.

That could have caused a problem.
Cyanogenmod 12.1 (Android 5.1.1)

What could be wrong?
Should I change the battery?

Only if you determine that the battery is bad. Don't depend on the phone's indication of the amount of charge - it's only showing you the terminal voltage of the battery. (The only other way of measuring state of charge would destroy the battery, so that's the only way we have.)

Voltage 4290 mV

What percentage charge is the phone showing at that voltage? What voltage is CPU-Z showing just before the battery fails? (Look at the Battery in Settings. Tap the graph to get the second page. And sudden drops to zero? If so, replace the battery unless it's obviously loose - in which case, put a piece of paper or two under it as a shim.)