# Properties

 Label 8-15e8-1.1-c1e4-0-1 Degree $8$ Conductor $2562890625$ Sign $1$ Analytic cond. $10.4192$ Root an. cond. $1.34038$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3·4-s − 3·9-s + 4·11-s + 4·16-s + 32·19-s − 2·29-s + 9·36-s − 10·41-s − 12·44-s − 5·49-s − 28·59-s − 14·61-s − 9·64-s + 8·71-s − 96·76-s − 12·79-s + 60·89-s − 12·99-s + 36·101-s − 20·109-s + 6·116-s + 26·121-s + 127-s + 131-s + 137-s + 139-s − 12·144-s + ⋯
 L(s)  = 1 − 3/2·4-s − 9-s + 1.20·11-s + 16-s + 7.34·19-s − 0.371·29-s + 3/2·36-s − 1.56·41-s − 1.80·44-s − 5/7·49-s − 3.64·59-s − 1.79·61-s − 9/8·64-s + 0.949·71-s − 11.0·76-s − 1.35·79-s + 6.35·89-s − 1.20·99-s + 3.58·101-s − 1.91·109-s + 0.557·116-s + 2.36·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 144-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$8$$ Conductor: $$3^{8} \cdot 5^{8}$$ Sign: $1$ Analytic conductor: $$10.4192$$ Root analytic conductor: $$1.34038$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{8} \cdot 5^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$1.313395972$$ $$L(\frac12)$$ $$\approx$$ $$1.313395972$$ $$L(\frac{3}{2})$$ 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)$
bad3$C_2^2$ $$1 + p T^{2} + p^{2} T^{4}$$
5 $$1$$
good2$C_2^3$ $$1 + 3 T^{2} + 5 T^{4} + 3 p^{2} T^{6} + p^{4} T^{8}$$
7$C_2^3$ $$1 + 5 T^{2} - 24 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8}$$
11$C_2^2$ $$( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2}$$
13$C_2^2$$\times$$C_2^2$ $$( 1 - T^{2} + p^{2} T^{4} )( 1 + 23 T^{2} + p^{2} T^{4} )$$
17$C_2^2$ $$( 1 - 18 T^{2} + p^{2} T^{4} )^{2}$$
19$C_2$ $$( 1 - 8 T + p T^{2} )^{4}$$
23$C_2^3$ $$1 + 37 T^{2} + 840 T^{4} + 37 p^{2} T^{6} + p^{4} T^{8}$$
29$C_2^2$ $$( 1 + T - 28 T^{2} + p T^{3} + p^{2} T^{4} )^{2}$$
31$C_2^2$ $$( 1 - p T^{2} + p^{2} T^{4} )^{2}$$
37$C_2^2$ $$( 1 - 58 T^{2} + p^{2} T^{4} )^{2}$$
41$C_2^2$ $$( 1 + 5 T - 16 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2}$$
43$C_2^2$$\times$$C_2^2$ $$( 1 - 61 T^{2} + p^{2} T^{4} )( 1 + 83 T^{2} + p^{2} T^{4} )$$
47$C_2^3$ $$1 + 45 T^{2} - 184 T^{4} + 45 p^{2} T^{6} + p^{4} T^{8}$$
53$C_2^2$ $$( 1 - 102 T^{2} + p^{2} T^{4} )^{2}$$
59$C_2^2$ $$( 1 + 14 T + 137 T^{2} + 14 p T^{3} + p^{2} T^{4} )^{2}$$
61$C_2^2$ $$( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} )^{2}$$
67$C_2^3$ $$1 + 125 T^{2} + 11136 T^{4} + 125 p^{2} T^{6} + p^{4} T^{8}$$
71$C_2$ $$( 1 - 2 T + p T^{2} )^{4}$$
73$C_2^2$ $$( 1 - 130 T^{2} + p^{2} T^{4} )^{2}$$
79$C_2^2$ $$( 1 + 6 T - 43 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2}$$
83$C_2^3$ $$1 + 85 T^{2} + 336 T^{4} + 85 p^{2} T^{6} + p^{4} T^{8}$$
89$C_2$ $$( 1 - 15 T + p T^{2} )^{4}$$
97$C_2^3$ $$1 + 190 T^{2} + 26691 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$