Consider the Dirichlet Series attached to the square of the Dedekind Eta function. Does this series admit an Euler product?

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If so, is it the product of two Dirichlet L-Functions? Or could it be a linear combination of products of two Dirichlet L-Functions? My guess is that it is definitely not the product of two, and that it is either the latter or completely independent of Dirichlet L-Functions.