Unfortunately, the answer is yes. Having a criminal conviction doesn't put someone in a protected class like race, gender, ethnicity, sexual orientation, disability, etc., does under the law. So a landlord is legally permitted to discriminate against someone based on their criminal history.
The California Fair Employment and Housing Act prevents landlords from discriminating between would-be tenants on the grounds of their sex, race, color, religion, sexual orientation, marital status, ancestry, national origin, source of income, disability or medical condition. However, in the case of Marina Point v. Wolfson, the California Supreme Court decided that the Unruh protections are not necessarily restricted to these characteristics.
However, the landlord must apply any standard evenly across the board in order for it to be legal discrimination. So if they deny an applicant who has a specific criminal conviction, then they have to deny any other applicant with the same background. Fair housing laws require you to apply background check and screening criteria consistently to every applicant. In other words, if you turn one would-be tenant away because of an entry on his criminal check but accept another ex-offender, you're breaking the law.
As an aside, in addition to being an attorney, I have also been a landlord for over 26 years...