Following Iwaniek and Kowalski, S5.10, page 130 we consider an angle character $\xi_k$ on the Gaussian integers $\mathbb Z[i]$ defined by $ \xi_k(\mathfrak a) = \left(\frac{\alpha}{|\alpha|}\right)^k $ where $k \equiv 0 \pmod 4$.

This gives an $L$-function $L(s,\xi_k)$ and a functional equation described by the following data: \begin{align*} L(s, \xi_k) &= \sum_{\mathfrak a} \xi_k(\mathfrak a) (N\mathfrak a)^{-s} \\ \gamma(s, \xi_k) &= \pi^{-s} \Gamma\left( s + \frac{\left\lvert k \right\rvert}{2} \right) \\ \Lambda(s, \xi_k) &= \pi^{-s} \Gamma\left( s + \frac{\left\lvert k \right\rvert}{2} \right) L(s, \xi_k) = \Lambda(1-s, \xi_k) \end{align*} Here $L(s,\xi_k)$ has degree $d=2$ and conductor $q=4$ (the discriminant of the Gaussian integers).

I&K had earlier written a prime number theorem $\psi(f,x) = rx - \frac{x^{\beta_f}}{\beta_f} + O\left(\sqrt{\mathfrak q(f)}x e^{ -\frac{c}{2d^4}\sqrt{\log x}} \right)$ which held for all $L$-functions, and apply this to deduce that $$ \sum_{|\pi| \le x} \left(\frac{\pi}{|\pi|}\right)^k = 4\delta_{k=0} \operatorname{Li}(x) + O\left(|k|xe^{-c\sqrt{\log x}}\right). $$

What I'm confused about is how the analytic conductor $\mathfrak q(\xi_k)$ was computed; the calculation seems to suggest that it is $O(|q|^2)$. But the definition of the analytic conductor earlier was $$ \mathfrak q(f) \overset{\text{def}}{=} q(f) \prod_{j=1}^d \left( |\kappa_j| + 3 \right)$$ but the $\gamma$ factor given above is not in the correct form to carry out this definition.

Is there something I'm missing? I'm guessing there might be some property of the $\Gamma$ function that transform $\gamma(s, \xi_k)$ to the right form but I'm not sure what it might be.

  • 3
    $\begingroup$ Use the duplication formula for the Gamma function which connects $\Gamma(2s)$ to $\Gamma(s) \Gamma(s+1/2)$. $\endgroup$
    – Lucia
    Jun 13 '15 at 1:00

Got it thanks to Lucia's comment. The main point is to use the duplication formula; we have $$ \Gamma\left(s+\frac{|k|}{2}\right) \cdot \sqrt\pi = 2^{s+\frac{|k|}{2}-1} \Gamma\left(\frac12s+\frac{|k|}{4}\right) \Gamma(\left(\frac12s+\frac{|k|}{4}+\frac12\right).$$ Since the conductor in our case is $q = 4$, this exactly knocks out the factor of $q^{-s/2}$ that is attached to the completed $L$-function. There is in fact an extra constant factor of $\pi^{-\frac12} 2^{\frac{|k|}{2}-1}$ floating around, but it has no effect on the results; the functional equation remains true and it goes away when we take the logarithmic derivative.

  • 1
    $\begingroup$ You must mean "Legendre duplication formula", or more simply "duplication formula", rather than "Lagrange interpolation formula". $\endgroup$
    – KConrad
    Jul 8 '18 at 14:51

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