# Properties

 Label 8-1840e4-1.1-c1e4-0-4 Degree $8$ Conductor $1.146\times 10^{13}$ Sign $1$ Analytic cond. $46599.3$ Root an. cond. $3.83307$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 2·3-s + 4·5-s + 3·7-s − 9-s − 4·11-s + 8·15-s − 17-s + 4·19-s + 6·21-s + 4·23-s + 10·25-s − 2·27-s + 19·29-s + 31-s − 8·33-s + 12·35-s − 3·37-s + 13·41-s + 6·43-s − 4·45-s − 6·47-s − 5·49-s − 2·51-s + 19·53-s − 16·55-s + 8·57-s − 23·59-s + ⋯
 L(s)  = 1 + 1.15·3-s + 1.78·5-s + 1.13·7-s − 1/3·9-s − 1.20·11-s + 2.06·15-s − 0.242·17-s + 0.917·19-s + 1.30·21-s + 0.834·23-s + 2·25-s − 0.384·27-s + 3.52·29-s + 0.179·31-s − 1.39·33-s + 2.02·35-s − 0.493·37-s + 2.03·41-s + 0.914·43-s − 0.596·45-s − 0.875·47-s − 5/7·49-s − 0.280·51-s + 2.60·53-s − 2.15·55-s + 1.05·57-s − 2.99·59-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{16} \cdot 5^{4} \cdot 23^{4}$$ Sign: $1$ Analytic conductor: $$46599.3$$ Root analytic conductor: $$3.83307$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 5^{4} \cdot 23^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$11.84673706$$ $$L(\frac12)$$ $$\approx$$ $$11.84673706$$ $$L(\frac{3}{2})$$ 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$C_1$ $$( 1 - T )^{4}$$
23$C_1$ $$( 1 - T )^{4}$$
good3$D_{4}$ $$( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} )^{2}$$
7$C_2 \wr C_2\wr C_2$ $$1 - 3 T + 2 p T^{2} - 11 T^{3} + 66 T^{4} - 11 p T^{5} + 2 p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}$$
11$C_2 \wr C_2\wr C_2$ $$1 + 4 T + 28 T^{2} + 92 T^{3} + 406 T^{4} + 92 p T^{5} + 28 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
13$D_4\times C_2$ $$1 + 11 T^{2} + 160 T^{4} + 11 p^{2} T^{6} + p^{4} T^{8}$$
17$C_2 \wr C_2\wr C_2$ $$1 + T + 50 T^{2} + 27 T^{3} + 1154 T^{4} + 27 p T^{5} + 50 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8}$$
19$C_2 \wr C_2\wr C_2$ $$1 - 4 T + 60 T^{2} - 188 T^{3} + 1590 T^{4} - 188 p T^{5} + 60 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
29$C_2 \wr C_2\wr C_2$ $$1 - 19 T + 233 T^{2} - 1922 T^{3} + 12034 T^{4} - 1922 p T^{5} + 233 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8}$$
31$C_2 \wr C_2\wr C_2$ $$1 - T + 23 T^{2} - 104 T^{3} + 1648 T^{4} - 104 p T^{5} + 23 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
37$C_2 \wr C_2\wr C_2$ $$1 + 3 T + 32 T^{2} + 349 T^{3} + 1638 T^{4} + 349 p T^{5} + 32 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
41$C_2 \wr C_2\wr C_2$ $$1 - 13 T + 209 T^{2} - 1602 T^{3} + 13682 T^{4} - 1602 p T^{5} + 209 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2 \wr C_2\wr C_2$ $$1 - 6 T + 136 T^{2} - 758 T^{3} + 8126 T^{4} - 758 p T^{5} + 136 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
47$C_2 \wr C_2\wr C_2$ $$1 + 6 T + 105 T^{2} + 298 T^{3} + 5324 T^{4} + 298 p T^{5} + 105 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
53$C_2 \wr C_2\wr C_2$ $$1 - 19 T + 178 T^{2} - 929 T^{3} + 4474 T^{4} - 929 p T^{5} + 178 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8}$$
59$C_2 \wr C_2\wr C_2$ $$1 + 23 T + 336 T^{2} + 3511 T^{3} + 29550 T^{4} + 3511 p T^{5} + 336 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2 \wr C_2\wr C_2$ $$1 + 188 T^{2} + 136 T^{3} + 15462 T^{4} + 136 p T^{5} + 188 p^{2} T^{6} + p^{4} T^{8}$$
67$C_2 \wr C_2\wr C_2$ $$1 - 3 T + 170 T^{2} - 391 T^{3} + 15834 T^{4} - 391 p T^{5} + 170 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}$$
71$C_2 \wr C_2\wr C_2$ $$1 - 3 T + 135 T^{2} - 104 T^{3} + 9080 T^{4} - 104 p T^{5} + 135 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2 \wr C_2\wr C_2$ $$1 + 32 T + 635 T^{2} + 8400 T^{3} + 83736 T^{4} + 8400 p T^{5} + 635 p^{2} T^{6} + 32 p^{3} T^{7} + p^{4} T^{8}$$
79$C_2 \wr C_2\wr C_2$ $$1 + 2 T + 176 T^{2} + 826 T^{3} + 15838 T^{4} + 826 p T^{5} + 176 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2 \wr C_2\wr C_2$ $$1 - 21 T + 428 T^{2} - 5005 T^{3} + 56054 T^{4} - 5005 p T^{5} + 428 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2 \wr C_2\wr C_2$ $$1 + 140 T^{2} - 1496 T^{3} + 6326 T^{4} - 1496 p T^{5} + 140 p^{2} T^{6} + p^{4} T^{8}$$
97$C_2 \wr C_2\wr C_2$ $$1 + 18 T + 460 T^{2} + 5038 T^{3} + 69350 T^{4} + 5038 p T^{5} + 460 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$