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For Hecke L-functions associated to a holomorphic cusp form $f$ of level $N$, the local factors can be decomposed into $$L_p(s, f) = (1-\lambda_f(p)p^{-s} + \chi_N(p)p^{-2s})^{-1}$$

where $\chi_N$ is the primitive character modulo $N$. Now take $\pi$ to be an automorphic representation for $GL_2$. What can be said about the local L-factor associated to $\pi$? Do we know the specific form in which it can be written? (something like the same but with $N$ being the depth, or the conductor of $\pi$?)

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  • $\begingroup$ The local $L$-factor of an automorphic representation is of the form $L_p(s,\pi) = (1-\lambda_{\pi_p}(p) p^{-s}+\omega_{\pi_p}(p) p^{-2s})^{-1}$, where $\pi_p$ is the local component of $\pi$ (i.e. a generic irreducible admissible representation of $\mathrm{GL}_2(\mathbb{Q}_p)$, and $\omega_{\pi_p}$ is the central character of $\pi_p$ (so is a character of $\mathbb{Q}_p^{\times}$. The product over $p$ and $\infty$ of these central characters is a unitary Hecke character $\omega_{\pi}$ of $\mathbb{Q} \backslash \mathbb{A_Q}^{\times}$, which is the idèlic lift of a Dirichlet character $\chi$. $\endgroup$
    – Peter Humphries
    Commented Oct 25, 2017 at 10:00
  • $\begingroup$ See Goldfeld and Hundley's book for further details. $\endgroup$
    – Peter Humphries
    Commented Oct 25, 2017 at 10:00
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    $\begingroup$ @PeterHumphries Thanks for the reference, I wasn't believing that the answer would be so "standard". So that means, comparing to the case of cusp forms I wrote, that $\chi_N$ is the central character of an automorphic representation associated to $f$ in some way? $\endgroup$
    – Gory
    Commented Oct 25, 2017 at 10:05
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    $\begingroup$ Of course both for cusp forms and automorphic representations, these are the local $L$-factors only at unramified primes. $\endgroup$
    – Will Sawin
    Commented Oct 25, 2017 at 10:06
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    $\begingroup$ @Gory As Peter said, at ramified primes $\lambda_{\pi_p}$ might be zero or nonzero, but the last term is always $0$ at ramified primes. $\endgroup$
    – Will Sawin
    Commented Oct 25, 2017 at 15:36

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