# Properties

 Label 8-980e4-1.1-c1e4-0-0 Degree $8$ Conductor $922368160000$ Sign $1$ Analytic cond. $3749.83$ Root an. cond. $2.79738$ 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 + 3·9-s − 6·11-s − 16·19-s + 5·25-s + 4·29-s + 4·31-s − 24·41-s + 12·45-s − 24·55-s − 20·59-s − 14·79-s + 9·81-s + 16·89-s − 64·95-s − 18·99-s + 24·101-s − 14·109-s + 31·121-s − 4·125-s + 127-s + 131-s + 137-s + 139-s + 16·145-s + 149-s + 151-s + ⋯
 L(s)  = 1 + 1.78·5-s + 9-s − 1.80·11-s − 3.67·19-s + 25-s + 0.742·29-s + 0.718·31-s − 3.74·41-s + 1.78·45-s − 3.23·55-s − 2.60·59-s − 1.57·79-s + 81-s + 1.69·89-s − 6.56·95-s − 1.80·99-s + 2.38·101-s − 1.34·109-s + 2.81·121-s − 0.357·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 1.32·145-s + 0.0819·149-s + 0.0813·151-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{4} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{8} \cdot 5^{4} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$3749.83$$ Root analytic conductor: $$2.79738$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{980} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{8} \cdot 5^{4} \cdot 7^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.01644324955$$ $$L(\frac12)$$ $$\approx$$ $$0.01644324955$$ $$L(\frac{3}{2})$$ 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 $$1$$
5$C_2^2$ $$1 - 4 T + 11 T^{2} - 4 p T^{3} + p^{2} T^{4}$$
7 $$1$$
good3$C_2$$\times$$C_2^2$ $$( 1 - p T^{2} )^{2}( 1 + p T^{2} + p^{2} T^{4} )$$
11$C_2^2$ $$( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}$$
13$C_2^2$ $$( 1 - 25 T^{2} + p^{2} T^{4} )^{2}$$
17$C_2^3$ $$1 + 9 T^{2} - 208 T^{4} + 9 p^{2} T^{6} + p^{4} T^{8}$$
19$C_2$ $$( 1 + T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2}$$
23$C_2^3$ $$1 + 42 T^{2} + 1235 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8}$$
29$C_2$ $$( 1 - T + p T^{2} )^{4}$$
31$C_2^2$ $$( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
37$C_2^2$$\times$$C_2^2$ $$( 1 - 73 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} )$$
41$C_2$ $$( 1 + 6 T + p T^{2} )^{4}$$
43$C_2^2$ $$( 1 - 70 T^{2} + p^{2} T^{4} )^{2}$$
47$C_2^3$ $$1 - 27 T^{2} - 1480 T^{4} - 27 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2^3$ $$1 + 70 T^{2} + 2091 T^{4} + 70 p^{2} T^{6} + p^{4} T^{8}$$
59$C_2^2$ $$( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2}$$
61$C_2^2$ $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
67$C_2^3$ $$1 + 34 T^{2} - 3333 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8}$$
71$C_2$ $$( 1 + p T^{2} )^{4}$$
73$C_2^2$$\times$$C_2^2$ $$( 1 - 97 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} )$$
79$C_2^2$ $$( 1 + 7 T - 30 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2}$$
83$C_2^2$ $$( 1 - 22 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2^2$ $$( 1 - 8 T - 25 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
97$C_2^2$ $$( 1 - 185 T^{2} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$