# Properties

 Degree $8$ Conductor $240100000000$ Sign $1$ Motivic weight $1$ Primitive no Self-dual yes Analytic rank $4$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s − 6·3-s + 2·4-s + 12·6-s − 8·7-s − 4·8-s + 15·9-s − 12·12-s + 16·14-s + 8·16-s − 30·18-s + 48·21-s − 2·23-s + 24·24-s − 18·27-s − 16·28-s − 16·29-s − 8·32-s + 30·36-s − 96·42-s + 8·43-s + 4·46-s − 30·47-s − 48·48-s + 34·49-s + 36·54-s + 32·56-s + ⋯
 L(s)  = 1 − 1.41·2-s − 3.46·3-s + 4-s + 4.89·6-s − 3.02·7-s − 1.41·8-s + 5·9-s − 3.46·12-s + 4.27·14-s + 2·16-s − 7.07·18-s + 10.4·21-s − 0.417·23-s + 4.89·24-s − 3.46·27-s − 3.02·28-s − 2.97·29-s − 1.41·32-s + 5·36-s − 14.8·42-s + 1.21·43-s + 0.589·46-s − 4.37·47-s − 6.92·48-s + 34/7·49-s + 4.89·54-s + 4.27·56-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \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^{8} \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: $$2^{8} \cdot 5^{8} \cdot 7^{4}$$ Sign: $1$ Motivic weight: $$1$$ Character: induced by $\chi_{700} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$4$$ Selberg data: $$(8,\ 2^{8} \cdot 5^{8} \cdot 7^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$=$$ $$0$$ $$L(\frac12)$$ $$=$$ $$0$$ $$L(\frac{3}{2})$$ 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$C_2^2$ $$1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4}$$
5 $$1$$
7$C_2$ $$( 1 + 4 T + p T^{2} )^{2}$$
good3$C_2$ $$( 1 + p T^{2} )^{2}( 1 + p T + p T^{2} )^{2}$$
11$C_2^3$ $$1 + 21 T^{2} + 320 T^{4} + 21 p^{2} T^{6} + p^{4} T^{8}$$
13$C_2^2$ $$( 1 + 14 T^{2} + p^{2} T^{4} )^{2}$$
17$C_2^3$ $$1 - 31 T^{2} + 672 T^{4} - 31 p^{2} T^{6} + p^{4} T^{8}$$
19$C_2^2$$\times$$C_2^2$ $$( 1 - 37 T^{2} + p^{2} T^{4} )( 1 + 26 T^{2} + p^{2} T^{4} )$$
23$C_2^2$ $$( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} )^{2}$$
29$C_2$ $$( 1 + 4 T + p T^{2} )^{4}$$
31$C_2^2$$\times$$C_2^2$ $$( 1 - 46 T^{2} + p^{2} T^{4} )( 1 - 13 T^{2} + p^{2} T^{4} )$$
37$C_2^3$ $$1 + 65 T^{2} + 2856 T^{4} + 65 p^{2} T^{6} + p^{4} T^{8}$$
41$C_2^2$ $$( 1 - 70 T^{2} + p^{2} T^{4} )^{2}$$
43$C_2$ $$( 1 - 2 T + p T^{2} )^{4}$$
47$C_2^2$ $$( 1 + 15 T + 122 T^{2} + 15 p T^{3} + p^{2} T^{4} )^{2}$$
53$C_2^3$ $$1 + 105 T^{2} + 8216 T^{4} + 105 p^{2} T^{6} + p^{4} T^{8}$$
59$C_2^3$ $$1 - 91 T^{2} + 4800 T^{4} - 91 p^{2} T^{6} + p^{4} T^{8}$$
61$C_2^2$ $$( 1 + 9 T + 88 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2}$$
67$C_2^2$ $$( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2}$$
71$C_2^2$ $$( 1 + 54 T^{2} + p^{2} T^{4} )^{2}$$
73$C_2^3$ $$1 - 71 T^{2} - 288 T^{4} - 71 p^{2} T^{6} + p^{4} T^{8}$$
79$C_2^3$ $$1 + 77 T^{2} - 312 T^{4} + 77 p^{2} T^{6} + p^{4} T^{8}$$
83$C_2^2$ $$( 1 + 26 T^{2} + p^{2} T^{4} )^{2}$$
89$C_2^2$ $$( 1 + 27 T + 332 T^{2} + 27 p T^{3} + p^{2} T^{4} )^{2}$$
97$C_2^2$ $$( 1 - 106 T^{2} + p^{2} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$