Am I Required to Give 60 Days' Notice to Move Out if It's Required By My Lease

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My question involves landlord-tenant law in the State of: CA

My lease states:

If this agreement is on a month-to-month basis, at least sixty (60) days' prior written notice must be given to Lessor if Lessee intends to vacate the premises.

Either party may terminate the Fixed Term lease on or after the expiration date by at least sixty (60) days' prior written notice.

If this Agreement is a fixed term lease, it shall automatically convert to a month-to-month basis upon the expiration date of the fixed term lease. In the absence of any written communication between the parties, the residency shall continue on a month-to-month basis including any changes, i.e. rent adjustments, having been made by Lessor with proper written notice.

My question is whether this is valid, or whether there's something in the Civil Code that would supersede it and only require 30 days' written notice.

Today is January 22; my lease ends on February 28. I was planning to give my 30-days' notice this week, but I found this clause in my lease, and I wanted to find out if I'm stuck paying their increased month-to-month rate for an additional 30 days.

I think this is the relevant section of the Code:
http://www.leginfo.ca.gov/cgi-bin/di...le=1940-1954.1

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In particular, I'm seeing this section:

1946. A hiring of real property, for a term not specified by the
parties, is deemed to be renewed as stated in Section 1945, at the
end of the term implied by law unless one of the parties gives
written notice to the other of his intention to terminate the same,
at least as long before the expiration thereof as the term of the
hiring itself, not exceeding 30 days; provided, however, that as to
tenancies from month to month either of the parties may terminate the
same by giving at least 30 days' written notice thereof at any time
and the rent shall be due and payable to and including the date of
termination. It shall be competent for the parties to provide by an
agreement at the time such tenancy is created that a notice of the
intention to terminate the same may be given at any time not less
than seven days before the expiration of the term thereof. The notice
herein required shall be given in the manner prescribed in Section
1162 of the Code of Civil Procedure or by sending a copy by certified
or registered mail addressed to the other party. In addition, the
lessee may give such notice by sending a copy by certified or
registered mail addressed to the agent of the lessor to whom the
lessee has paid the rent for the month prior to the date of such
notice or by delivering a copy to the agent personally.

That sounds to me like I can give 30 days' notice, but I'm not sure if that's a right that I contracted away by signing their standard lease form.