# Properties

 Label 8-168e4-1.1-c1e4-0-6 Degree $8$ Conductor $796594176$ Sign $1$ Analytic cond. $3.23851$ Root an. cond. $1.15822$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 6·3-s − 2·4-s − 6·5-s − 2·7-s + 21·9-s + 6·11-s − 12·12-s − 36·15-s + 12·20-s − 12·21-s + 17·25-s + 54·27-s + 4·28-s + 36·29-s − 30·31-s + 36·33-s + 12·35-s − 42·36-s − 12·44-s − 126·45-s + 7·49-s + 6·53-s − 36·55-s − 18·59-s + 72·60-s − 42·63-s + 8·64-s + ⋯
 L(s)  = 1 + 3.46·3-s − 4-s − 2.68·5-s − 0.755·7-s + 7·9-s + 1.80·11-s − 3.46·12-s − 9.29·15-s + 2.68·20-s − 2.61·21-s + 17/5·25-s + 10.3·27-s + 0.755·28-s + 6.68·29-s − 5.38·31-s + 6.26·33-s + 2.02·35-s − 7·36-s − 1.80·44-s − 18.7·45-s + 49-s + 0.824·53-s − 4.85·55-s − 2.34·59-s + 9.29·60-s − 5.29·63-s + 64-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \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^{12} \cdot 3^{4} \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: $$2^{12} \cdot 3^{4} \cdot 7^{4}$$ Sign: $1$ Analytic conductor: $$3.23851$$ Root analytic conductor: $$1.15822$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{12} \cdot 3^{4} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.370212305$$ $$L(\frac12)$$ $$\approx$$ $$2.370212305$$ $$L(\frac{3}{2})$$ 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$C_2^2$ $$1 + p T^{2} + p^{2} T^{4}$$
3$C_2$ $$( 1 - p T + p T^{2} )^{2}$$
7$C_2^2$ $$1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4}$$
good5$C_2^2$$\times$$C_2^2$ $$( 1 + 2 T^{2} + p^{2} T^{4} )( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} )$$
11$C_2^2$$\times$$C_2^2$ $$( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 - 10 T^{2} + p^{2} T^{4} )$$
13$C_2$ $$( 1 + p T^{2} )^{4}$$
17$C_2^2$ $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
19$C_2^2$ $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
23$C_2^2$ $$( 1 + p T^{2} + p^{2} T^{4} )^{2}$$
29$C_2^2$ $$( 1 - 18 T + 137 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2}$$
31$C_2$$\times$$C_2^2$ $$( 1 + 10 T + p T^{2} )^{2}( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} )$$
37$C_2^2$ $$( 1 + p T^{2} + p^{2} T^{4} )^{2}$$
41$C_2$ $$( 1 + p T^{2} )^{4}$$
43$C_2$ $$( 1 - p T^{2} )^{4}$$
47$C_2^2$ $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
53$C_2^2$$\times$$C_2^2$ $$( 1 - 6 T + 65 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 - 94 T^{2} + p^{2} T^{4} )$$
59$C_2^2$$\times$$C_2^2$ $$( 1 - 10 T^{2} + p^{2} T^{4} )( 1 + 18 T + 167 T^{2} + 18 p T^{3} + p^{2} T^{4} )$$
61$C_2^2$ $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
67$C_2^2$ $$( 1 + p T^{2} + p^{2} T^{4} )^{2}$$
71$C_2$ $$( 1 - p T^{2} )^{4}$$
73$C_2^2$$\times$$C_2^2$ $$( 1 - 14 T + 123 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 123 T^{2} + 14 p T^{3} + p^{2} T^{4} )$$
79$C_2$$\times$$C_2^2$ $$( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + 21 T^{2} + 10 p T^{3} + p^{2} T^{4} )$$
83$C_2^2$$\times$$C_2^2$ $$( 1 - 30 T + 383 T^{2} - 30 p T^{3} + p^{2} T^{4} )( 1 + 30 T + 383 T^{2} + 30 p T^{3} + p^{2} T^{4} )$$
89$C_2^2$ $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
97$C_2^2$$\times$$C_2^2$ $$( 1 - 2 T - 93 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$