# Properties

 Label 8-1560e4-1.1-c1e4-0-6 Degree $8$ Conductor $5.922\times 10^{12}$ Sign $1$ Analytic cond. $24077.2$ Root an. cond. $3.52939$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·3-s + 10·9-s + 6·13-s + 10·17-s − 2·23-s − 2·25-s + 20·27-s − 8·29-s + 24·39-s + 20·43-s + 15·49-s + 40·51-s − 2·53-s − 2·61-s − 8·69-s − 8·75-s − 10·79-s + 35·81-s − 32·87-s + 8·101-s + 24·103-s + 10·107-s − 16·113-s + 60·117-s + 35·121-s + 127-s + 80·129-s + ⋯
 L(s)  = 1 + 2.30·3-s + 10/3·9-s + 1.66·13-s + 2.42·17-s − 0.417·23-s − 2/5·25-s + 3.84·27-s − 1.48·29-s + 3.84·39-s + 3.04·43-s + 15/7·49-s + 5.60·51-s − 0.274·53-s − 0.256·61-s − 0.963·69-s − 0.923·75-s − 1.12·79-s + 35/9·81-s − 3.43·87-s + 0.796·101-s + 2.36·103-s + 0.966·107-s − 1.50·113-s + 5.54·117-s + 3.18·121-s + 0.0887·127-s + 7.04·129-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{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 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}$$ Sign: $1$ Analytic conductor: $$24077.2$$ Root analytic conductor: $$3.52939$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{12} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$16.98958283$$ $$L(\frac12)$$ $$\approx$$ $$16.98958283$$ $$L(\frac{3}{2})$$ 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 $$1$$
3$C_1$ $$( 1 - T )^{4}$$
5$C_2$ $$( 1 + T^{2} )^{2}$$
13$C_2^2$ $$1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4}$$
good7$D_4\times C_2$ $$1 - 15 T^{2} + 116 T^{4} - 15 p^{2} T^{6} + p^{4} T^{8}$$
11$D_4\times C_2$ $$1 - 35 T^{2} + 544 T^{4} - 35 p^{2} T^{6} + p^{4} T^{8}$$
17$D_{4}$ $$( 1 - 5 T + 36 T^{2} - 5 p T^{3} + p^{2} T^{4} )^{2}$$
19$D_4\times C_2$ $$1 - 40 T^{2} + 1054 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8}$$
23$D_{4}$ $$( 1 + T + 42 T^{2} + p T^{3} + p^{2} T^{4} )^{2}$$
29$C_2$ $$( 1 + 2 T + p T^{2} )^{4}$$
31$C_2^2$ $$( 1 - 58 T^{2} + p^{2} T^{4} )^{2}$$
37$D_4\times C_2$ $$1 - 115 T^{2} + 5836 T^{4} - 115 p^{2} T^{6} + p^{4} T^{8}$$
41$D_4\times C_2$ $$1 - 87 T^{2} + 5216 T^{4} - 87 p^{2} T^{6} + p^{4} T^{8}$$
43$D_{4}$ $$( 1 - 10 T + 94 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2}$$
47$C_2$ $$( 1 - p T^{2} )^{4}$$
53$D_{4}$ $$( 1 + T + 68 T^{2} + p T^{3} + p^{2} T^{4} )^{2}$$
59$D_4\times C_2$ $$1 - 92 T^{2} + 7990 T^{4} - 92 p^{2} T^{6} + p^{4} T^{8}$$
61$D_{4}$ $$( 1 + T + 84 T^{2} + p T^{3} + p^{2} T^{4} )^{2}$$
67$D_4\times C_2$ $$1 + 56 T^{2} + 4254 T^{4} + 56 p^{2} T^{6} + p^{4} T^{8}$$
71$D_4\times C_2$ $$1 - 163 T^{2} + 15768 T^{4} - 163 p^{2} T^{6} + p^{4} T^{8}$$
73$D_4\times C_2$ $$1 - 208 T^{2} + 19774 T^{4} - 208 p^{2} T^{6} + p^{4} T^{8}$$
79$D_{4}$ $$( 1 + 5 T + 126 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 296 T^{2} + 35614 T^{4} - 296 p^{2} T^{6} + p^{4} T^{8}$$
89$D_4\times C_2$ $$1 - 335 T^{2} + 43792 T^{4} - 335 p^{2} T^{6} + p^{4} T^{8}$$
97$D_4\times C_2$ $$1 - 379 T^{2} + 54724 T^{4} - 379 p^{2} T^{6} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$