Special values of Dirichlet series and zeta integrals

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For f and g polynomials in p variables, we relate the special value at a non-positive integer s=-N, obtained by analytic continuation of the Dirichlet series ζ(s;f,g)=∑ₖ₁₌₀∞... ∑ₖₚ₌₀∞ g(k₁,...,kₚ)f(k₁,...,kₚ)⁻ˢ((s)≫0), to special values of zeta integrals Z(s;f,g)=∫ₓ∈[₀,∞₎ₚ g(x)f(x)⁻ˢdx ((s)≫0). We prove a simple relation between ζ(-N;f,g) and Z(-N;fₐ,gₐ), where for a∈ ᵖ,fₐ(x) is the shifted polynomial fₐ(x)=f(a+x). By direct calculation we prove the product rule for zeta integrals at s=0, degree(fh)· Z(0;fh,g)=degree(f)· Z(0;f,g)+degree(h)· Z(0;h,g), and deduce the corresponding rule for Dirichlet series at s=0, degree(fh)·ζ(0;fh,g)=degree(f) ·ζ(0;f,g)+degree(h)·ζ(0;h,g). This last formula generalizes work of Shintani and Chen-Eie.