Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Converse theorems assuming a partial Euler product   Associated to a newform $f(z)$ is a Dirichlet series $L_f(s)$ with functional
equation and Euler product. Hecke showed that if the Dirichlet series $F(s)$
has a functional equation of a particular form, then $F(s)=L_f(s)$ for some
holomorphic newform $f(z)$ on $\Gamma(1)$. Weil extended this result to
$\Gamma_0(N)$ under an assumption on the twists of $F(s)$ by Dirichlet
characters. Conrey and Farmer extended Hecke's result for certain small $N$,
assuming that the local factors in the Euler product of $F(s)$ were of a
special form. We make the same assumption on the Euler product and describe an
approach to the converse theorem using certain additional assumptions. Some of
the assumptions may be related to second order modular forms.

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The categories of this paper are: quant-ph
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The categories of this paper are: mtrl-th
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The categories of this paper are: patt-sol
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The categories of this paper are: nucl-th
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The categories of this paper are: solv-int
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The categories of this paper are: math-ph
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The categories of this paper are: supr-con
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The categories of this paper are: funct-an
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The categories of this paper are: dg-ga
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The categories of this paper are: chao-dyn
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The categories of this paper are: hep-lat
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The categories of this paper are: gr-qc
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The categories of this paper are: hep-th
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The categories of this paper are: adap-org
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The categories of this paper are: cs
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The categories of this paper are: econ