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.



What is the analytic conductor of this Hecke L-function?
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.


