# Properties

 Degree $8$ Conductor $1.216\times 10^{12}$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4-s + 9-s + 10·11-s − 14·19-s + 4·31-s + 36-s + 20·41-s + 10·44-s − 2·49-s + 8·59-s − 8·61-s − 64-s + 8·71-s − 14·76-s − 24·79-s + 28·89-s + 10·99-s − 4·109-s + 47·121-s + 4·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + ⋯
 L(s)  = 1 + 1/2·4-s + 1/3·9-s + 3.01·11-s − 3.21·19-s + 0.718·31-s + 1/6·36-s + 3.12·41-s + 1.50·44-s − 2/7·49-s + 1.04·59-s − 1.02·61-s − 1/8·64-s + 0.949·71-s − 1.60·76-s − 2.70·79-s + 2.96·89-s + 1.00·99-s − 0.383·109-s + 4.27·121-s + 0.359·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \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^{4} \cdot 3^{4} \cdot 5^{8} \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: $$2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{1050} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$4.956585065$$ $$L(\frac12)$$ $$\approx$$ $$4.956585065$$ $$L(\frac{3}{2})$$ 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$C_2^2$ $$1 - T^{2} + T^{4}$$
3$C_2^2$ $$1 - T^{2} + T^{4}$$
5 $$1$$
7$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
good11$C_2^2$ $$( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2}$$
13$C_2^2$ $$( 1 - T^{2} + p^{2} T^{4} )^{2}$$
17$C_2^3$ $$1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8}$$
19$C_2$ $$( 1 - T + p T^{2} )^{2}( 1 + 8 T + p T^{2} )^{2}$$
23$C_2^3$ $$1 + 45 T^{2} + 1496 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8}$$
29$C_2$ $$( 1 + p T^{2} )^{4}$$
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$ $$( 1 + 26 T^{2} + p^{2} T^{4} )( 1 + 47 T^{2} + p^{2} T^{4} )$$
41$C_2$ $$( 1 - 5 T + p T^{2} )^{4}$$
43$C_2^2$ $$( 1 + 58 T^{2} + p^{2} T^{4} )^{2}$$
47$C_2^3$ $$1 - 27 T^{2} - 1480 T^{4} - 27 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2^3$ $$1 + 25 T^{2} - 2184 T^{4} + 25 p^{2} T^{6} + p^{4} T^{8}$$
59$C_2^2$ $$( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2}$$
61$C_2^2$ $$( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2}$$
67$C_2^3$ $$1 - 10 T^{2} - 4389 T^{4} - 10 p^{2} T^{6} + p^{4} T^{8}$$
71$C_2$ $$( 1 - 2 T + p T^{2} )^{4}$$
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$C_2^2$ $$( 1 + 12 T + 65 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2}$$
83$C_2^2$ $$( 1 - 22 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2^2$ $$( 1 - 14 T + 107 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{2}$$
97$C_2$ $$( 1 - 18 T + p T^{2} )^{2}( 1 + 18 T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$