In a large bowl, stir together the mayonnaise, lemon zest, paprika, celery seeds, salt, black pepper and cayenne pepper. Slather the chicken inside and out with mayonnaise mixture, place it in the bowl and let stand in the refrigerator while you prepare the grill.
Prepare a charcoal or gas grill for indirect grilling. If using a gas grill, preheat the grill to medium and place a drip pan in the center of the grill. If using a charcoal grill, light the coals. When they are red hot, use tongs to move them into two piles on opposite sides of the grill. Let coals burn until they are covered with a thin layer of gray ash. Set a metal drip pan in the center of the grill between the piles of coals.
When the grill is ready, open the beer can and drink a few sips (or discard). Put the can in the chicken's rear cavity, keeping the can upright. Stand the chicken upright on the grill grate over the drip pan. Cover the grill and cook for 1 1/2- 2 hours or until the chicken is very tender and nearly falling off the bone. (Using charcoal, you will need to add 10 to 12 fresh charcoal briquettes per side after 1 hour.) When done, the internal temperature of the chicken should register 180° on a meat thermometer inserted in the thickest part of the thigh but not touching the bone.
Transfer the chicken to the cutting board or serving platter, taking care not to spill the hot beer. Let the chicken stand for 5 minutes; remove and discard the beer and carve the chicken.