Product of integers in list without libraries

Let's say I'm not allowed to use libraries. How do I go about calculating the product of indexes in a list. Let's assume none of the integers are 0 or less. The problem gets harder as I'm trying to calculate the indexes vertically.

bigList = [[1, 2, 3, 4, 5],
           [1, 2, 3, 4, 5],
           [1, 2, 3, 4, 5],
           [1, 2, 3, 4, 5],
           [1, 2, 3, 4, 5]]

With numpy the solution for my problem would be:

import numpy as np   
print([np.prod(l) for l in zip(*bigList)])

[1, 32, 243, 1024, 3125]

However without it my solution is much more chaotic:

rotateY = [l for l in zip(*bigList)]
productList = [1]* len(bigList)
count = 0
for l in rotateY:
    for i in l:
        productList[count] *= i
    count += 1
print(productList)

[1, 32, 243, 1024, 3125]