If you don't set an expiry date, the cookie will be a 'session cookie' so will be deleted when the browser session ends.

Is this a correct syntax if I remove the expiration?

setcookie("Cookie_Name", $login_id, , "/", ".domain.com");

No, this will throw a PHP error. Use 0 instead of the normal expiry time--this will set the cookie as a "session" cookie that will expire whenever the user closes his browser or leaves it inactive for a period of time--usually 20-30 minutes.

Remember that the PHP manual [php.net] usually answers such questions, in this instance with the documentation for setting a cookie [php.net]