Archit's Challenge 3

Algebra Level 5

Given

$f(x) = x^{13} + 2x^{12} + 3x^{11} + 4x^{10} + \cdots + 13x + 14,$

denote

$N = f(a) \times f\left(a^2\right) \times f\left(a^3\right) \times\cdots \times f\left(a^{14} \right),$

where $a = \cos\left( \frac{2\pi}{15} \right) + i \sin\left( \frac{2\pi}{15} \right) .$

Then what is the value of $M$ such that $N^\frac{1}{M} = 15 ?$