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Let $K$ be a field and let $S=\bigoplus_{n\geq 0} S_n$ be a positively graded$K$-algebra. Given $M=\bigoplus_{n\geq 0} M_n$, a finitely generated graded$S$-module, and $w>0$, we introduce the function $\zeta_M(z,w):=\sum_{n=0}^{\infty}\frac{H(M,n)}{(n+w)^z}$, where $H(M,n):=\dim_K M_n$, $n\geq0$, is the Hilbert function of $M$, and we study the relations between thealgebraic properties of $M$ and the analytic properties of $\zeta_M(z,w)$. Inparticular, in the standard graded case, we prove that the multiplicity of $M$,$e(M)=(m-1)!\lim_{w\searrow 0}Res_{z=m}\zeta_M(z,w)$.

On a Zeta-Barnes type function associated to graded modules