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I also think it is the NFC antenna.
I concur it is NFC see here https://stellatech.com/sony-c2105-xperia-l-antenna-module-nfc,a,en,814851.html
You can carefully remove the NFC antenna from the old cover and put in the new one, I did it so and it works.
hahah.. you know guys, to me sony is just one of those companies who treat their customers bad or not worth once their product are bought. I'm from India and I have an xperia L and it was working well. Then one day i found an problem with the wireless adapter of the phone which i wasnt able to use the bluetooth and wifi. So as expected, I went to the service center to get my phone fixed. AND HERE IS WHAT HAPPENED !!! The guy in the service center didnt find a way to open the back door of my phone. (I wonder how sony gave such authorisation to these service center). With the hope that my phone will be fixed, I gave my phone to the guy and went as the guy said it'll take another 10 days to get the phone fixed.
Later after 10 days, i got my phone! wait !!! A BROKEN PHONE!! Yes sony users and sony, MY PHONE WAS BROKEN!! I found a no of cracks in the back door pannel and the chrome lining of the phone. When i claimed that my phone wasnt damaged when I gave it to him, he simply replied "We cant do anything about it ! If you want a new door pannel you can get a new one for 700 rs and if you do warranty will be void"
Sony!! is this how you make money ?? or havent you trained your service center personnels properly??
Hearing those ruthless words from the guy and feeling devasted for paying such a great deal of money for this phone. I hardly spoke a word and i left with agony, any second more i'd have stayed there, I'd have thrown my XPERIA AWAY IN THE SERVICE CENTER