Customers can not log in into Magento with Chrome browser, the login page just reloads without any errors or warnings.
But when customer logins in other browsers and admin back-end login function properly.

Cause
Magento sets an extra session cookie that overrides new cookies set by new sessions, as a result, session expires right after login.
Navigate to System > Configuration > Web > Session Cookie Management and set the following settings:

Cookie Lifetime: 7200
Cookie Path: /
Cookie Domain:
example.com
Use HTTP Only: Yes
Cookie restriction mode: No
Cookie Path: /
Cookie Domain:
example.com
Use HTTP Only: Yes
Cookie restriction mode: No
- After this, Flush Magento cache and delete the domain cookies in the Chrome browser. Refer to the above link to check how to
clear cookies :
https://support.google.com/chrome/answer/95647?hl=en- Refresh/reload magento customer login page and try to log in as usual.