Geodesic
On the zeta-functions of systems of quadratic forms
Matematičeskie zametki, Tome 20 (1976) no. 1, pp. 27-34.

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We prove that the zeta-function of $m$ variables of a system of $m$ positive-definite quadratic forms has a meromorphic analytic continuation onto the whole $m$-dimensional complex space. 
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     title = {On the zeta-functions of systems of quadratic forms},
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A. N. Andrianov. On the zeta-functions of systems of quadratic forms. Matematičeskie zametki, Tome 20 (1976) no. 1, pp. 27-34. http://geodesic.mathdoc.fr/item/MZM_1976_20_1_a3/