A simple remark on Dirichlet series

Let L(s) be an ordinary Dirichlet series absolutely convergent in some right halfplane which has meromorphic continuation to the entire complex plane. Let chi be a Dirichlet character. Then one might often expect that the twist L (s, x) of L (s) by chi has the same analytic properties. For instance,as is well known this in general can be proved if L(s) is attached to an automorphic form and so in addition one knows that L (s) is of finite order and when completed with appropriate F-factors satisfies a functional equation, see for example [2] in the case of GL(2) and [1] in the context of Koecher-Maass series on Sp(n). In the present paper we would like to give explicit evidence that the above expectation is not always satisfied automatically. In fact, we will give an example of a Dirichlet series with rational coefficients which is absolutely convergent for Re (s) > 1, extends to an entire function and such that there is a character twist which has analytic continuation to the entire plane except for an essential singularity at s = 1. To exclude any misunderstandings, let us point out that our function neither is of finite order nor satisfies a functional equation of appropriate type and so in particular does not satisfy the conditions of any converse theorem.