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The Langlands-Shahidi method says that the $L$-functions of automorphic representations appear in the constant terms of Eisenstein series. Since those Eisenstein series have analytic continuation and functional equation, the $L$-function also inherits those nice properties.

Most expositions of the Langlands-Shahidi method are phrased in the adelic language, which I am not too comfortable with. So I would like to understand it in a more classical language, without using adeles.

My question is: Suppose you have a weight $2$ eigenform $f \in S_2(\Gamma_0(N))$. Consider the $L$-function $L(f,s)$ attached to $f$. Is there a reference that explains the Langlands-Shahidi method for $L(f,s)$ without referring to adeles? Namely, in classical language, what is the Eisenstein series whose constant term contains the $L$-function $L(f,s)$? How can the nice analytic properties of $L(f,s)$ be read off from the analytic properties of the Eisenstein series?

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    $\begingroup$ Note that adeles were introduced to make life easier, not harder. $\endgroup$
    – GH from MO
    Commented Jan 14, 2023 at 20:49
  • $\begingroup$ The $L$-functions you see in the constant terms of GL(2) Eisenstein series are all from GL(1): the Riemann zeta function and Dirichlet $L$-functions. To see $L(f,s)$ in the constant term of an Eisenstein series, you would need to look at GL(3). The classical theory is much harder to develop here. It might be in Bump's Springer Lecture Notes. But the adelic/representation theoretic approach is much more developed. $\endgroup$
    – Stopple
    Commented Jan 14, 2023 at 22:22

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Well, as part of a constant term of an Eisenstein series, it would appear in the constant term of an Eisenstein series attached to the 2,1 parabolic (or 1,2...) in GL3, with the cuspform on the GL2 factor of the Levi component. My relatively recent book(s) with Cambridge U Press... also available on-line... treat some aspects of this, in both classical and adelic terms.

But also you should look at Langlands' SLN 544, and Goldfeld/Goldfeld-Hundley's books on GLn...

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