Need to verify strange behavior with Product[]

Question

I'm running 12.1.0.0 on MacOS X x86 (64-bit).  The expression
Product[1 - 1/n^5, {n, 2, Infinity}]

yields 0 as output, which is clearly false.  Wolfram Alpha gives the correct result.  In fact, it appears that
Product[1 - 1/n^k, {n, 2, Infinity}]

for any odd integer $k ge 5$ returns 0.  Replacement of Infinity with a large positive integer gives the expected result.  Is anyone else able to replicate this issue?  I have a strong suspicion that this is due to some problem with handling the choice of complex branch.

A.Z. · Answer

This is an answer as I cannot comment yet.

Is anyone else able to replicate this issue?

Yes, I can confirm that Mathematica 12.1.0.0 on Linux and Mac gives the answer 0, same for the odd integers greater than 5.
In Mathematica 11.3.0.0 the code
Product[1 - 1/n^5, {n, 2, Infinity}] // InputForm

yields
1/(Gamma[2 + (-1)^(1/5)]*Gamma[2 - (-1)^(2/5)]*Gamma[2 + (-1)^(3/5)]*Gamma[2 - (-1)^(4/5)])

Odd integers bigger than 5 also seem to work.

Need to verify strange behavior with Product[]

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