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You didn't mention what year, but given that the photo is of a pre-facelift XJS instrument cluster, and you said it is a convertible, that would mean it is between '89-'91 (or '89-'90 if it was old in CA) and almost certainly a V12. Therefore, it won't have a check engine light, nor would it have an OBDII port. So it has to be smog checked the old fashioned way with tailpipe sniffer.
If running properly, and all catalytic converters are in place and working, if should easily pass.