H25B EXPERIENCES #1 ENG FUEL ANNUNCIATOR. RUN CHECKLIST; DECLARE EMERGENCY AND DIVERT TO NEAREST SUITABLE ARPT. ANNUNCIATOR GOES OUT DURING ARRIVAL.

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WE HAD A NORMAL TAXI AND DEP FROM ZZZ1 USING RWY XX. WE WERE TURNED N ALMOST IMMEDIATELY AND BEFORE TOO LONG WERE GIVEN DIRECT ABC. WE CROSSED ABC; WITH INSTRUCTIONS TO CLB TO FL370. I WAS THE PF AND AS WE WERE CLBING THROUGH FL340 WE EXPERIENCED AN AMBER LOOK-UP LIGHT AND A #1 ENG FUEL ANNUNCIATOR. OUR IMMEDIATE ACTION WAS TO CONSULT THE CHKLIST WHICH INSTRUCTED US TO: 'PULL THE PWR LEVER TO THE AFFECTED ENG BACK UNTIL THE LIGHT WENT OUT OR UNTIL REACHING IDLE PWR.' AS I SLOWLY PULLED THE L PWR LEVER TOWARDS IDLE; THE FO ASKED ATC FOR A LEVELOFF AT FL350. ONCE THE L ENG PWR LEVER WAS AT IDLE; THE LIGHT REMAINED ILLUMINATED. THE NEXT ITEM ON THE CHKLIST WAS A NOTE THAT STATED: 'SHOULD THE LIGHT REMAIN ILLUMINATED AND THE ENG BEGIN TO MALFUNCTION; INITIATE THE ENG SHUTDOWN CHKLIST.' SINCE THE LIGHT WAS REMAINING ILLUMINATED; I DECIDED THAT IT WOULD BE IN THE BEST INTEREST OF SAFETY FOR BOTH THE PAX AND THE ACFT TO DIVERT TO AN AIRFIELD CLOSE BY AND LAND. AT THAT MOMENT WE WERE JUST A FEW MI W OF THE ABC2 VOR AND ZZZ SEEMED OUR BETTER OPTION. THE FO INFORMED ATC OF OUR INTENTION TO DIVERT AND WHEN ASKED BY ATC FOR THE REASON HE STATED THAT WE WERE DIVERTING FOR MECHANICAL REASONS. ATC THEN ASKED IF WE WERE DECLARING AN EMER. I INSTRUCTED THE FO TO REPLY AFFIRMATIVE AS I WAS UNSURE AS TO WHAT ACTUAL PROB THE ENG WAS EXPERIENCING. ATC IMMEDIATELY TURNED US TOWARDS ZZZ AND INSTRUCTED US TO DSND TO 10000 FT. I TURNED THE ACFT TOWARDS ZZZ AND RETARDED THE R PWR LEVER TO START OUR DSCNT. THE LIGHT REMAINED ILLUMINATED AND PWR INDICATIONS SEEMED NORMAL WITH BOTH LEVERS AT IDLE; UNTIL WE REACHED AROUND FL290/280; AT WHICH TIME THE #1 ENG FUEL ANNUNCIATOR EXTINGUISHED. THE #1 ENG FUEL ANNUNCIATOR REMAINED EXTINGUISHED FOR THE REMAINDER OF THE FLT. A FEW MINS AFTER THE LIGHT WENT OUT; WE MADE AN ATC FREQ CHANGE; THE NEW CTLR ASKED IF WE WERE STILL IN AN EMER CONDITION. AFTER OBSERVING THAT THE #1 ENG FUEL ANNUNCIATOR HAD REMAINED EXTINGUISHED FOR SEVERAL MINS; I TOLD THE FO THAT WE COULD PROBABLY CANCEL THE EMER; BUT THAT I THOUGHT THAT A PRECAUTIONARY LNDG WAS STILL A GOOD IDEA. ATC INFORMED US THAT THEY HAD TO CONTINUE TO TREAT US AS AN EMER; BUT THAT WE WOULD NO LONGER RECEIVER PRIORITY HANDLING.