I don't believe your stamp is a fake.
Here is an L-5TS

which you may compare to the L-5; big difference.
/Users/rkdesign/Desktop/L-6 HS California.jpg

Hoover T-7 | similar L-03 E except dim d 13.5 mm to 14.0 mm | at least 2 devices are
included in this listing

Your image closely resembles an L-10 HS (Minus the integral added company information.)
L-10 INTEGRAL HS

(description L-10)
often perfinned & not
well printed | large caps between bars
11.0 mm apart | most commonly
applied over Buro prints | used
1938/12 and 1939/01
California is not my strongest area In precancels. I suggest contacting the Precancel Stamp Society:
Michael Hynes
P.O. Box 866
Folsom, CA 95763
Email: mike@precancels.com
He'll get you directed to the correct CALIF specialists.
Good Luck,
Hal Klein
(Yes, I'm a PSS Member.)