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Generally you do.
But not all J pickups sound the same. - It depends upon the resistance of the coil [or coils if it is a humbucker internally], which way they are wired if there are two [as you also noticed in your new pickup], the magnet material and structure, and a bunch of other factors.
Also, too many people judge a J pickup by how the one usually located next to the bridge sounds.
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Yes, it's pretty without a doubt !

- Maybe you should sell it to me.
And hum IS a dealbreaker. - I hacked up the body of my brand new 1974 Ric 4001 rather early on, to try other pickups, to get rid of the hum, because I still liked its neck so much.
The coil resistance as you described it did exactly what it should, obeying Ohm's Law.
Based upon what you said, that pickup has two 6K coils.
In series they will add, making 12K.
In parallel, they will divide by the number of coils [if identical to each other], making 3K.
I suggest that you shop for strings next, and another set to consider is GHS Brite Flats, which are ground flat like D'Addario Half Rounds, but I like the GHS better, especially for short scale, as they are not as light of a gauge in short [and medium] scale, which will match a short scale bass better and not flop around with inconsistent pitch.
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