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Check condition and age of the battery. Many of the electrical 'faults' and 'gremlins' chased on these forums can be traced to a failing or weak battery.
Battery should have a minimum of 12.6 volts across the terminals when tested with a voltmeter ten minutes after the engine is turned off. If it has less than 12.6 volts, try charging it at 5 to 10 amps overnight and retest. If the battery still has less than 12.6 volts, it should be replaced.
Batteries really matter on these cars. My fog lights would come on when I started the car or the side lights would come on and a few other surprises. Just replaced the light switch and that solved all of that. I did have the battery checked first, however.
Minimal route is 8 miles but it is a local commute car. Never a battrey issue but I'll check the volts.
Not sure of the exact number, but years ago I was taught it requires about 20 minutes of freeway driving just to replenish the battery after a routine engine start. Any less than that repeated over and over will slowly run down the battery.
Scroll down to the battery section here for details on adding a battery tender:
There have been many threads on S-types and batteries/battery voltages over on the British Jaguar Forum. It would seem that the charging system on the V6 version was originally designed for conventional lead/acid batteries and tech has moved on with the replacement of antimony in the lead plate alloy with calcium. Consequently, the "smart" charging system can reportedly only partially charge the currently available crop of batteries up to a 75-80% of maximum charge level.
There is much discussion on this and one is contained in the following thread from roughly post #55 on:
Those voltages are all well below the 13.8V from the alternator so charging will take place.
All these car batteries gradually lose the ability to take charge, however, and that is sped up by being left discharged (if that occurs).
Also, quite a number of cars develop a power drain which typically goes unnoticed for ages, and it's easy to blame the battery or alternator - wrongly.