How big can i print this photo?

Its dimensions are 4608 x 3072
Taken with a Nikon D3100 at 55mm
f/8
1/250… I want to get it printed atleast 11 x 14 but if I could go bigger I will.
What's the biggest I can go without it looking bad?
p.s. (I'm entering it into a contest so I want it to look as good as possible)

Let's go all the way and look at printing. In general we consider 300 pixels per inch sufficient for a high quality print.So, divide 4608 by 300 and we get 15, 4, do the same for 3072 and we get 10. That's not a standard sized print but 11x14 is and so you could do that with excellent quality. Even a 13x19 print would still be very nice.

I agree with the math above, using 300 ppi as your goal for best quality, but with a high-quality image, such as you are getting with your D3100, you can go considerably lower. At just under 200 ppi, it works out to 24x16, which is a multiple of the 3x2 format from your camera, so there would be no cropping involved. It then follows that you could easily get any size smaller than that with no problem. I dare say, if the image is worthy, you would probably be satisfied all the way up to 36x24, if you don't do any cropping. The reason this works is because the viewing distance for a print of that size is more like 3-4 feet and so you would not notice the very minute pixelation if it occurred at all.

Go here and follow my directions for doing a test of your image before ordering a large print:

You can print it any size you want. It's a 14 megapixel image!

You could print it billboard size if you want without it looking bad.

Because the bigger you print images, the larger the viewing distance, therefore the less dpi you need.

300dpi (dots per inch) is considered magazine quality so there shouldn't be any problem with making an 11 x 14 print from your image file. However… Any imperfections, no matter how small, will be glaringly apparent at such a large print size. If the focus is the tiniest bit off and barely noticeable on a 4 x 6 print it will be objectionable at 11 x 14 because an 11 x 14 print is 6.42 times larger than a 4 x 6 print. (11x14 = 154 square inches; 4x6 = 24 square inches. 154/24 = 6.42)