# Properties

 Degree $8$ Conductor $3.842\times 10^{12}$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 5·9-s − 6·11-s + 2·19-s − 8·29-s + 10·31-s − 8·41-s − 2·49-s + 30·59-s + 10·61-s + 2·79-s + 9·81-s + 14·89-s + 30·99-s − 6·101-s − 10·109-s + 31·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 20·169-s − 10·171-s + ⋯
 L(s)  = 1 − 5/3·9-s − 1.80·11-s + 0.458·19-s − 1.48·29-s + 1.79·31-s − 1.24·41-s − 2/7·49-s + 3.90·59-s + 1.28·61-s + 0.225·79-s + 81-s + 1.48·89-s + 3.01·99-s − 0.597·101-s − 0.957·109-s + 2.81·121-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 + 0.0783·163-s + 0.0773·167-s − 1.53·169-s − 0.764·171-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \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^{12} \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: $$2^{12} \cdot 5^{8} \cdot 7^{4}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{1400} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{12} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.3245539403$$ $$L(\frac12)$$ $$\approx$$ $$0.3245539403$$ $$L(\frac{3}{2})$$ 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 $$1$$
7$C_2^2$ $$1 + 2 T^{2} + p^{2} T^{4}$$
good3$C_2^3$ $$1 + 5 T^{2} + 16 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8}$$
11$C_2^2$ $$( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}$$
13$C_2$ $$( 1 - 4 T + p T^{2} )^{2}( 1 + 4 T + p T^{2} )^{2}$$
17$C_2^3$ $$1 + 9 T^{2} - 208 T^{4} + 9 p^{2} T^{6} + p^{4} T^{8}$$
19$C_2$ $$( 1 - 8 T + p T^{2} )^{2}( 1 + 7 T + p T^{2} )^{2}$$
23$C_2^3$ $$1 - 3 T^{2} - 520 T^{4} - 3 p^{2} T^{6} + p^{4} T^{8}$$
29$C_2$ $$( 1 + 2 T + p T^{2} )^{4}$$
31$C_2^2$ $$( 1 - 5 T - 6 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2}$$
37$C_2^3$ $$1 + 65 T^{2} + 2856 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8}$$
41$C_2$ $$( 1 + 2 T + p T^{2} )^{4}$$
43$C_2^2$ $$( 1 - 70 T^{2} + p^{2} T^{4} )^{2}$$
47$C_2^3$ $$1 + 69 T^{2} + 2552 T^{4} + 69 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2^3$ $$1 + 105 T^{2} + 8216 T^{4} + 105 p^{2} T^{6} + p^{4} T^{8}$$
59$C_2^2$ $$( 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2}$$
61$C_2^2$ $$( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2}$$
67$C_2^3$ $$1 + 53 T^{2} - 1680 T^{4} + 53 p^{2} T^{6} + p^{4} T^{8}$$
71$C_2$ $$( 1 + p T^{2} )^{4}$$
73$C_2^2$$\times$$C_2^2$ $$( 1 - 46 T^{2} + p^{2} T^{4} )( 1 + 143 T^{2} + p^{2} T^{4} )$$
79$C_2^2$ $$( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} )^{2}$$
83$C_2^2$ $$( 1 - 22 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2^2$ $$( 1 - 7 T - 40 T^{2} - 7 p T^{3} + p^{2} T^{4} )^{2}$$
97$C_2^2$ $$( 1 - 190 T^{2} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$