# Properties

 Label 2-2e6-4.3-c8-0-8 Degree $2$ Conductor $64$ Sign $1$ Analytic cond. $26.0722$ Root an. cond. $5.10609$ Motivic weight $8$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

# Origins

## Dirichlet series

 L(s)  = 1 + 1.05e3·5-s + 6.56e3·9-s + 478·13-s − 6.33e4·17-s + 7.20e5·25-s + 1.40e6·29-s − 9.25e5·37-s + 3.57e6·41-s + 6.91e6·45-s + 5.76e6·49-s + 9.62e6·53-s − 2.07e7·61-s + 5.03e5·65-s − 5.47e7·73-s + 4.30e7·81-s − 6.67e7·85-s − 3.02e7·89-s − 1.73e8·97-s − 1.45e8·101-s + 1.94e8·109-s + 2.80e8·113-s + 3.13e6·117-s + ⋯
 L(s)  = 1 + 1.68·5-s + 9-s + 0.0167·13-s − 0.758·17-s + 1.84·25-s + 1.99·29-s − 0.494·37-s + 1.26·41-s + 1.68·45-s + 49-s + 1.21·53-s − 1.49·61-s + 0.0282·65-s − 1.92·73-s + 81-s − 1.27·85-s − 0.482·89-s − 1.95·97-s − 1.39·101-s + 1.37·109-s + 1.72·113-s + 0.0167·117-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$64$$    =    $$2^{6}$$ Sign: $1$ Analytic conductor: $$26.0722$$ Root analytic conductor: $$5.10609$$ Motivic weight: $$8$$ Rational: yes Arithmetic: yes Character: $\chi_{64} (63, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 64,\ (\ :4),\ 1)$$

## Particular Values

 $$L(\frac{9}{2})$$ $$\approx$$ $$2.899080654$$ $$L(\frac12)$$ $$\approx$$ $$2.899080654$$ $$L(5)$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
good3 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
5 $$1 - 1054 T + p^{8} T^{2}$$
7 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
11 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
13 $$1 - 478 T + p^{8} T^{2}$$
17 $$1 + 63358 T + p^{8} T^{2}$$
19 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
23 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
29 $$1 - 1407838 T + p^{8} T^{2}$$
31 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
37 $$1 + 925922 T + p^{8} T^{2}$$
41 $$1 - 3577922 T + p^{8} T^{2}$$
43 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
47 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
53 $$1 - 9620638 T + p^{8} T^{2}$$
59 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
61 $$1 + 20722082 T + p^{8} T^{2}$$
67 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
71 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
73 $$1 + 54717118 T + p^{8} T^{2}$$
79 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
83 $$( 1 - p^{4} T )( 1 + p^{4} T )$$
89 $$1 + 30265918 T + p^{8} T^{2}$$
97 $$1 + 173379838 T + p^{8} T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$