Mash the butter with the grated garlic, oregano, coriander, red pepper flakes, lemon zest, orange zest, 1 teaspoon salt and a few grinds of pepper in a bowl. Spoon onto a piece of plastic wrap and shape into a log. Wrap and refrigerate until firm, at least 1 hour.
Position a rack in the lowest part of the oven, remove the other racks and preheat to 350 degrees F. Put the turkey neck, onion, garlic halves, carrot and celery in a large roasting pan; set a rack on top. Add 2 cups water.
Pat the turkey dry. Slice the butter log into 1/2-inch-thick rounds. Gently separate the turkey skin from the breast, the thighs and the top of the drumsticks. Slide the butter under the skin. Stuff the lemon halves and orange half inside the cavity. Tie the legs together with kitchen twine. Season the turkey generously with salt and pepper and place breast-side up on the rack; tuck the wings under the body. Let sit at room temperature 30 minutes.
Roast the turkey 1 1/2 hours, then baste with the pan drippings. Continue to roast, basting every 30 minutes, until the skin is crisp and golden and a thermometer inserted into the thigh registers 160 degrees F to 165 degrees F, 45 minutes to 1 1/2 hours more. Transfer the turkey (on the rack) to a rimmed baking sheet and let rest 30 minutes before carving. Use the pan drippings for gravy.
Prepare the garlic-citrus butter (step 1) up to 2 days ahead and refrigerate.