Finding the number of analytic functions which vanish only on a given set.

Let $S = \{0\}\cup \{\frac{1}{4n+7} : n =1,2\ldots\}$. Just how to find the variety of analytic features which disappear just on $S$?

Alternatives are

a: $\infty$

b: $0$

c: $1$

d: $2$