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It sounds like you are missing the internet settings. You could try to download them automatically via Settings > More... > Internet settings. Make sure that you are not connected to a Wi-Fi network while you download them.
If the download fail or that it still does not work, I suggest that you contact your operator to make sure that you have the correct settings and/or that you try to add them manually via Settings > More... > Mobile networks > Access Point Names.
More info can be found at the following link:
http://userguide.sonymobile.com/referrer.php?region=global-en&product=xperia-z2#Internet-and-messagi...
