the canonical product
Weierstrass's factorization let you choose the degrees p_n of the elementary factors however you liked, as long as the product converged. That freedom is a little unsatisfying — different choices give different-looking products for the same zeros. The canonical product removes the slack: it is the most economical product over a given zero sequence, formed by using the SAME smallest degree p for every elementary factor, the smallest p that still makes the product converge.
Concretely, given zeros a_1, a_2, ... with |a_n| -> infinity, look for the smallest non-negative integer p such that sum 1/|a_n|^(p+1) converges. (This p exists whenever any such product is possible.) The canonical product is then the product over n of E_p(z/a_n), with that single fixed degree p used throughout. Because every factor has the same degree, the construction is canonical — determined by the zeros alone, with no arbitrary choices left. It is the cleanest entire function with exactly those zeros, the natural representative of its class.
The canonical product matters because it is the bridge between zeros and growth. Hadamard's theorem says that for an entire function of finite order, the e^(g(z)) factor in the Weierstrass form is forced to be the exponential of a polynomial, and the elementary factors collapse to a canonical product — so the function is essentially determined by its zeros and a polynomial. The smallest degree p that makes the canonical product converge is tightly linked to how fast the function can grow. So the canonical product is not a mere tidying-up; it is where the construction theory meets the growth theory of entire functions.
For zeros at the integers n != 0, sum 1/|n|^(p+1) converges first at p = 1 (sum 1/n^2 < infinity) but diverges at p = 0 (sum 1/n = infinity). So the canonical product uses degree p = 1 throughout: product over n != 0 of E_1(z/n), which equals the product over n >= 1 of (1 - z^2/n^2). This is exactly the product that builds sin(pi z) / (pi z).
Pick the smallest p with sum 1/|a_n|^(p+1) finite; for the integers that is p = 1, giving the sine product.
The canonical product is determined by the zeros alone, but it is not the whole entire function — you still multiply by some e^(g(z)). What is canonical is the product part; the unit factor remains free unless growth conditions (Hadamard's theorem) constrain g to a polynomial.