# Properties

 Label 8-896e4-1.1-c1e4-0-10 Degree $8$ Conductor $644513529856$ Sign $1$ Analytic cond. $2620.23$ Root an. cond. $2.67480$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 4·5-s + 8·7-s + 4·9-s − 8·11-s − 12·13-s − 4·25-s − 32·35-s + 24·43-s − 16·45-s + 34·49-s + 32·55-s + 12·61-s + 32·63-s + 48·65-s − 24·67-s − 64·77-s + 6·81-s − 96·91-s − 32·99-s + 4·101-s + 48·103-s + 8·107-s − 24·113-s − 48·117-s + 20·121-s + 44·125-s + 127-s + ⋯
 L(s)  = 1 − 1.78·5-s + 3.02·7-s + 4/3·9-s − 2.41·11-s − 3.32·13-s − 4/5·25-s − 5.40·35-s + 3.65·43-s − 2.38·45-s + 34/7·49-s + 4.31·55-s + 1.53·61-s + 4.03·63-s + 5.95·65-s − 2.93·67-s − 7.29·77-s + 2/3·81-s − 10.0·91-s − 3.21·99-s + 0.398·101-s + 4.72·103-s + 0.773·107-s − 2.25·113-s − 4.43·117-s + 1.81·121-s + 3.93·125-s + 0.0887·127-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{28} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$2620.23$$ Root analytic conductor: $$2.67480$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{896} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{28} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.737250394$$ $$L(\frac12)$$ $$\approx$$ $$1.737250394$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
7$C_2$ $$( 1 - 4 T + p T^{2} )^{2}$$
good3$D_4\times C_2$ $$1 - 4 T^{2} + 10 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8}$$
5$D_{4}$ $$( 1 + 2 T + 8 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2}$$
11$D_{4}$ $$( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
13$D_{4}$ $$( 1 + 6 T + 32 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
17$C_2^2$ $$( 1 - 18 T^{2} + p^{2} T^{4} )^{2}$$
19$D_4\times C_2$ $$1 - 52 T^{2} + 1290 T^{4} - 52 p^{2} T^{6} + p^{4} T^{8}$$
23$D_4\times C_2$ $$1 - 60 T^{2} + 1766 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8}$$
29$C_2^2$ $$( 1 - 10 T^{2} + p^{2} T^{4} )^{2}$$
31$C_2^2$ $$( 1 + 14 T^{2} + p^{2} T^{4} )^{2}$$
37$C_2^2$ $$( 1 - 58 T^{2} + p^{2} T^{4} )^{2}$$
41$D_4\times C_2$ $$1 - 36 T^{2} + 614 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8}$$
43$D_{4}$ $$( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} )^{2}$$
47$C_2^2$ $$( 1 + 46 T^{2} + p^{2} T^{4} )^{2}$$
53$C_2^2$ $$( 1 - 58 T^{2} + p^{2} T^{4} )^{2}$$
59$D_4\times C_2$ $$1 - 20 T^{2} - 1686 T^{4} - 20 p^{2} T^{6} + p^{4} T^{8}$$
61$D_{4}$ $$( 1 - 6 T + 56 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
67$D_{4}$ $$( 1 + 12 T + 158 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
71$D_4\times C_2$ $$1 - 132 T^{2} + 11366 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8}$$
73$C_2^2$ $$( 1 - 98 T^{2} + p^{2} T^{4} )^{2}$$
79$C_2^2$ $$( 1 - 146 T^{2} + p^{2} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 164 T^{2} + 19530 T^{4} - 164 p^{2} T^{6} + p^{4} T^{8}$$
89$D_4\times C_2$ $$1 - 132 T^{2} + 7910 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8}$$
97$C_2^2$ $$( 1 - 50 T^{2} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$