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Let $f(z)=\sum_{n\ge 1}a(n)n^{(k-1/2)/2}e(nz)\in S_{k+1/2}(\Gamma_0(4N),\chi)$ be a cuspidal Hecke eigenform. Let $$M(s)=\sum_{p\ge 2, \text{prime}}\frac{|a(p)|^2}{p^s}$$ and $$R_f(s)=\sum_{n\ge 1}\frac{|a(n)|^2}{n^s}$$ Is there a functional equation link the two series ?

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    $\begingroup$ The second is approximately the Rankin-Selberg convolution of the standard $L$-function attached to $f$ with itself. It has a functional equation. The first displayed expression is a fragment of the logarithm of the second, and I'd wager that it has a natural boundary... $\endgroup$
    – paul garrett
    Commented Aug 12, 2016 at 21:50
  • $\begingroup$ @paulgarrett, Thank you very much for your comment. $\endgroup$
    – Med
    Commented Aug 12, 2016 at 22:03
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    $\begingroup$ Dear @paulgarrett, Could you please explain to me what you mean by "The first displayed expression is a fragment of the logarithm of the second," $\endgroup$
    – Med
    Commented Aug 12, 2016 at 22:47
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    $\begingroup$ Hecke eigenform means $R_f(s)$ has an Euler product $R_f(s)= \prod_p 1+\sum_{k=1}^\infty a(p^k)^2 p^{-sk}$, so that $\log R_f(s) = \mathcal{O}(1) + \sum_p a(p)^2 p^{-s}+ a(p^2)^2 p^{-2s}$ for $Re(s) > k+1/2+\epsilon$ (or something like that) $\endgroup$
    – reuns
    Commented Aug 13, 2016 at 2:39
  • $\begingroup$ @paulgarrett , @ user1952009, I think that you are overlooked that f is a modular form of half-integral weight ....!!! $\endgroup$
    – Med
    Commented Aug 13, 2016 at 17:12

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