Evaluate $\lim_{n\to\infty}\prod_{k=0}^n\cos{\frac{a}{p^k}}$ where $p\in\mathbb{N}$

Question

Evaluate $$\lim_{n\to\infty}\prod_{k=1}^n\cos{\dfrac{a}{p^k}}, p\in\mathbb{N}$$I can use double angle formulae to evaluate $$\lim_{n\to\infty}\prod_{k=1}^n\cos{\dfrac{a}{2^k}}=\dfrac{\sin a}{a}$$ But I wonder the evaluation when $p=3,4,5\cdots$ $$$$The question might be complex-analysis-related.