Masonry steel nails are available in 1-1/4”, 1-1/2”, and 2”, used for attaching the strap to the wall for the downspout.      
Masonry Nail
2” screw shank zinc plated steel nails are used to attach the gutter to the house through the hanger.  The screw shank nail can also be used under the shingles to attach the coil strap to the roof decking for slanted fascia or extra support.  
Screw Shank Nails
A 2” ring shank copper nail used for copper guttering installation.  
Copper Ring Shank
A #10, ¼” blue plastic anchor is used in mortar before inserting a drive pin, for a downspout strap.  
Blue Plastic Anchor
A ¼” x 1-1/4” drive pin is used to attach a downspout strap.  
Drive Pin