My 05 STR's engine light has been illuminated for several weeks and the error code came up as P1646. With this code, there seems to be two primary causes A) an issue with the oxygen sensor's internal heating circuit or B) an issue with the voltage being sent from the ECM which controls the sensor's heating circuit at Bank 1, Sensor 1 (...possibly blown fuse, short in or bad/loose wiring loom).
Bank 1 = RH side (U.S Passenger-Side)
Sensor 1 = Upstream sensor (closest to the engine)

IMPORTANT NOTE:
While searching on the web, I found several instances stating Bank 1 is the Driver-Side and this is not the case for a U.S. 05 Jag STR.
Bank 1 = RH side (U.S. Passenger-Side)
Bank 2 = LH side (U.S. Driver-Side)

I did the following to diagnose the problem before installing a new sensor:

1. Purchased new sensor (Denso p/n 234-9030).
2. Put vehicle on jack stands and removed engine's lower plastic cover.
3. Disconnected battery.
4. Used 10mm socket and extension to remove driver and passenger O2 sensor harnesses.
5. Disconnected both upstream sensors from their harnesses.
6. Used Multi-meter to test wires from ECM side to ensure voltage was present at engine startup. If no voltage is present at startup, check for blown fuse, short in or bad/loose wiring loom in the wiring from ECM's drive circuit (Bank 1, Sensor 1 & Bank 2, Sensor 1).
7. Used Multi-meter to verify sensor's wires had continuity on heating circuit (usually two black wires). If they don't have continuity (A.K.A. open circuit) the sensor's heater circuit is defective and the sensor needs replaced. My Bank 1, Sensor 1 sensor measured as an open circuit.
8. Used PB Blaster and an open-end 7/8" wrench to remove and replace the defective sensor. Used anti-size on the threads of the new sensor.
9. Used 10mm socket and extension to replace driver and passenger O2 sensor harnesses.
10. Reconnected battery.
11. Started engine, went under vehicle, listened for exhaust leaks, double-checked all sensors were correctly installed and tightly clamped.
12. Cleared engine code.
13. Restarted engine twice and success! No more engine light with the error code P1646!
14. Replaced lower plastic engine cover and removed vehicle from jack stands.
15. Complete.

Please help by jumping in and listing the fuse box location and numbers of the ECM's driver circuits for Bank 1, Sensor 1 & Bank 2, Sensor 2. THANKS!

 

See handbook & JTIS for fuses, relays etc. Free downloads if you don't already have them.

Realistically the PCM (aka ECM) is the very last suspect.