cannot login with integrated google account in mobile web mode
At the moment the mobile version of the login screen does not allow the use of Google Authentication.
An account that is created through Gmail, does not have a password attached to it, so when you enter that email and password, it will not work.
If you sign into your account on the Web (non mobile), click on the "gear" icon at the top right and go to Settings, then Account. You will be able to attach a password to your account. You will then be able to sign in through the mobile version of the site.