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Let $\pi$ be an automorphic form of GL(n)/$\mathbb{Q}$ with standard $L$-function $$L(s,\pi)=\prod_p \prod_{i=1}^n(1-\frac{\alpha_{p,i}}{p^s})^{-1},$$ where $\{\alpha_{p,i}:i=1,\dots,n\}$ are the Satake parameters at $p$.

  1. What's the Euler product of Asai L-function of $\pi$ in terms of $\{\alpha_{p,i}:i=1,\dots,n\}$ in the simplest non-trivial case?

  2. What do we know about Asai L-function? Functional equations or poles?

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    $\begingroup$ Should that be $p^s$, not $p^2$, in the denominator? Otherwise your $L(s, \pi)$ is independent of $s$! More importantly, the Asai $L$-function is (AFAIK) associated to a finite extension of global fields $L / K$ and an automorphic representation $\pi$ of $GL_n / L$, corresponding under Langlands to the "tensor induction" map $GL_n / L \to GL_{nd} / K$ where $d = [L : K]$; if $L = K$ it reduces to the standard $L$-function of $\pi$. So if $L = \mathbf{Q}$ there is no difference between the Asai and standard $L$-functions. $\endgroup$
    – David Loeffler
    May 29, 2014 at 8:28
  • $\begingroup$ If you like my answer, please accept it officially (so that it turns green). Thanks in advance! $\endgroup$
    – GH from MO
    Aug 19, 2018 at 23:13

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I think your questions are answered in Asai's original paper (Math. Ann. 226 (1977), 81-94). Theorem 1 (on page 86) describes holomorphicity, poles, and the functional equation. Theorem 2 (on page 87) describes the Euler product.

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