Normalized local factors for the product formula ($|x{|}_{\mathfrak{p}}=(N\mathfrak{p}{)}^{-{\mathrm{ord}}_{\mathfrak{p}}(x)}$)

The product formula uses finite-place factors given by ideal norms, the ordinary modulus at real embeddings, and the modulus squared at complex places. The complex square accounts for the pair of conjugate embeddings. Equivalently ordinary complex modulus can be used with weight two in the product. These normalizations cancel the absolute global field norm.