the genus of a canonical product
/ genus: JEE-nus /
When you form the canonical product over a sequence of zeros, you must pick a single degree p for all the elementary factors — the smallest one that makes the product converge. That integer p has a name: it is the genus of the canonical product (and, more broadly, part of the genus of the entire function). It is a single whole number that records how the elementary factors had to be tuned, and hence, indirectly, how densely the zeros are packed.
Precisely, p is the smallest non-negative integer with sum 1/|a_n|^(p+1) < infinity. The more slowly the zeros run to infinity — the denser they are — the larger p must be to force convergence, so the genus measures the 'thickness' of the zero set. The full genus of an entire function f, in the Hadamard sense, is the maximum of this p and the degree of the polynomial sitting in the exponent g(z) = (a polynomial) of its factorization. So genus combines two pieces of data: how packed the zeros are (through p) and how big the unit factor's polynomial is.
Genus is the down-to-earth integer cousin of the (possibly fractional) order of an entire function. They are tightly related: for an entire function of finite order, the genus h and the order rho satisfy h <= rho <= h + 1 — the genus pins the order down to one of two adjacent integers' worth of room. So if you can read off the genus from the factorization, you immediately bracket the growth rate. This is why genus, an artefact of how you had to build the product, turns out to carry real analytic information about how fast the function grows.
The sine product has zeros at the integers, where the smallest p with sum 1/n^(p+1) < infinity is p = 1, so its canonical product has genus 1. Its exponent polynomial is degree 0 (the e^g factor is constant), so the genus of sin(pi z) is max(1, 0) = 1. Its order is rho = 1, consistent with the bound 1 <= 1 <= 2.
For sin(pi z): canonical-product genus 1, constant unit factor, total genus 1 — matching its order 1.
Genus is always an integer; order need not be (order can be any non-negative real, even infinite). Do not confuse the two — genus brackets the order via h <= rho <= h + 1, but they are not equal in general.