# Properties

 Label 8-4650e4-1.1-c1e4-0-5 Degree $8$ Conductor $4.675\times 10^{14}$ Sign $1$ Analytic cond. $1.90072\times 10^{6}$ Root an. cond. $6.09347$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·4-s − 2·9-s + 2·11-s + 3·16-s + 6·19-s − 12·29-s + 4·31-s + 4·36-s + 12·41-s − 4·44-s + 19·49-s + 36·59-s + 24·61-s − 4·64-s − 6·71-s − 12·76-s + 42·79-s + 3·81-s + 30·89-s − 4·99-s + 6·101-s + 28·109-s + 24·116-s − 33·121-s − 8·124-s + 127-s + 131-s + ⋯
 L(s)  = 1 − 4-s − 2/3·9-s + 0.603·11-s + 3/4·16-s + 1.37·19-s − 2.22·29-s + 0.718·31-s + 2/3·36-s + 1.87·41-s − 0.603·44-s + 19/7·49-s + 4.68·59-s + 3.07·61-s − 1/2·64-s − 0.712·71-s − 1.37·76-s + 4.72·79-s + 1/3·81-s + 3.17·89-s − 0.402·99-s + 0.597·101-s + 2.68·109-s + 2.22·116-s − 3·121-s − 0.718·124-s + 0.0887·127-s + 0.0873·131-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 31^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 31^{4}$$ Sign: $1$ Analytic conductor: $$1.90072\times 10^{6}$$ Root analytic conductor: $$6.09347$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 31^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$7.460033390$$ $$L(\frac12)$$ $$\approx$$ $$7.460033390$$ $$L(\frac{3}{2})$$ 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$C_2$ $$( 1 + T^{2} )^{2}$$
3$C_2$ $$( 1 + T^{2} )^{2}$$
5 $$1$$
31$C_1$ $$( 1 - T )^{4}$$
good7$D_4\times C_2$ $$1 - 19 T^{2} + 184 T^{4} - 19 p^{2} T^{6} + p^{4} T^{8}$$
11$D_{4}$ $$( 1 - T + 18 T^{2} - p T^{3} + p^{2} T^{4} )^{2}$$
13$C_2^2$ $$( 1 - 22 T^{2} + p^{2} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 - 32 T^{2} + 766 T^{4} - 32 p^{2} T^{6} + p^{4} T^{8}$$
19$D_{4}$ $$( 1 - 3 T + 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 - 59 T^{2} + 1720 T^{4} - 59 p^{2} T^{6} + p^{4} T^{8}$$
29$D_{4}$ $$( 1 + 6 T + 50 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
37$D_4\times C_2$ $$1 - 112 T^{2} + 5806 T^{4} - 112 p^{2} T^{6} + p^{4} T^{8}$$
41$D_{4}$ $$( 1 - 6 T + 74 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2}$$
43$D_4\times C_2$ $$1 + 53 T^{2} + 1744 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8}$$
47$D_4\times C_2$ $$1 - 152 T^{2} + 10126 T^{4} - 152 p^{2} T^{6} + p^{4} T^{8}$$
53$D_4\times C_2$ $$1 - 23 T^{2} + 4216 T^{4} - 23 p^{2} T^{6} + p^{4} T^{8}$$
59$D_{4}$ $$( 1 - 18 T + 182 T^{2} - 18 p T^{3} + p^{2} T^{4} )^{2}$$
61$C_2$ $$( 1 - 6 T + p T^{2} )^{4}$$
67$D_4\times C_2$ $$1 + 56 T^{2} + 4254 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8}$$
71$D_{4}$ $$( 1 + 3 T + 106 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 - 79 T^{2} + 12112 T^{4} - 79 p^{2} T^{6} + p^{4} T^{8}$$
79$D_{4}$ $$( 1 - 21 T + 230 T^{2} - 21 p T^{3} + p^{2} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 124 T^{2} + 7830 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8}$$
89$D_{4}$ $$( 1 - 15 T + 196 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2}$$
97$C_2^2$ $$( 1 - 158 T^{2} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$