Making use of the classical techniques [cf. E. Landau and A. Walfisz, Rend. Palermo 44, 82-86 (1920) and Estermann, Proc. Lond. Math. Soc. 27, 435-448 (1928)] the author proves that the Euler product \[ \prod_{\chi (p)=1}(1-p^{-s})^{-1}\quad\text{(and therefore the product }\prod_{\chi (p)=-1}(1-p^{-s})^{-1}) \] can be analytically continued as a meromorphic function of s to the half-plane Re s\(>0\) but has the line Re s\(=0\) as its natural boundary for analytic continuation. Here p ranges over all the rational primes, \(\chi\) denotes a quadratic Dirichlet character.