# Properties

 Label 8-882e4-1.1-c3e4-0-5 Degree $8$ Conductor $605165749776$ Sign $1$ Analytic cond. $7.33396\times 10^{6}$ Root an. cond. $7.21385$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 4·2-s + 4·4-s − 5·5-s − 16·8-s − 20·10-s + 67·11-s − 82·13-s − 64·16-s + 92·17-s + 43·19-s − 20·20-s + 268·22-s + 148·23-s − 80·25-s − 328·26-s − 154·29-s − 520·31-s − 64·32-s + 368·34-s − 7·37-s + 172·38-s + 80·40-s − 852·41-s − 214·43-s + 268·44-s + 592·46-s − 576·47-s + ⋯
 L(s)  = 1 + 1.41·2-s + 1/2·4-s − 0.447·5-s − 0.707·8-s − 0.632·10-s + 1.83·11-s − 1.74·13-s − 16-s + 1.31·17-s + 0.519·19-s − 0.223·20-s + 2.59·22-s + 1.34·23-s − 0.639·25-s − 2.47·26-s − 0.986·29-s − 3.01·31-s − 0.353·32-s + 1.85·34-s − 0.0311·37-s + 0.734·38-s + 0.316·40-s − 3.24·41-s − 0.758·43-s + 0.918·44-s + 1.89·46-s − 1.78·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{4} \cdot 3^{8} \cdot 7^{8}$$ Sign: $1$ Analytic conductor: $$7.33396\times 10^{6}$$ Root analytic conductor: $$7.21385$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{882} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{4} \cdot 3^{8} \cdot 7^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.5914546154$$ $$L(\frac12)$$ $$\approx$$ $$0.5914546154$$ $$L(\frac{5}{2})$$ 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$C_2$ $$( 1 - p T + p^{2} T^{2} )^{2}$$
3 $$1$$
7 $$1$$
good5$D_4\times C_2$ $$1 + p T + 21 p T^{2} - 66 p^{2} T^{3} - 494 p^{2} T^{4} - 66 p^{5} T^{5} + 21 p^{7} T^{6} + p^{10} T^{7} + p^{12} T^{8}$$
11$D_4\times C_2$ $$1 - 67 T + 1041 T^{2} - 52662 T^{3} + 4142284 T^{4} - 52662 p^{3} T^{5} + 1041 p^{6} T^{6} - 67 p^{9} T^{7} + p^{12} T^{8}$$
13$D_{4}$ $$( 1 + 41 T + 4478 T^{2} + 41 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 - 92 T + 1902 T^{2} + 17664 p T^{3} - 78413 p^{2} T^{4} + 17664 p^{4} T^{5} + 1902 p^{6} T^{6} - 92 p^{9} T^{7} + p^{12} T^{8}$$
19$D_4\times C_2$ $$1 - 43 T - 9305 T^{2} + 110252 T^{3} + 64683544 T^{4} + 110252 p^{3} T^{5} - 9305 p^{6} T^{6} - 43 p^{9} T^{7} + p^{12} T^{8}$$
23$D_4\times C_2$ $$1 - 148 T - 2526 T^{2} - 14208 T^{3} + 182283043 T^{4} - 14208 p^{3} T^{5} - 2526 p^{6} T^{6} - 148 p^{9} T^{7} + p^{12} T^{8}$$
29$D_{4}$ $$( 1 + 77 T + 9574 T^{2} + 77 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
31$D_4\times C_2$ $$1 + 520 T + 144563 T^{2} + 34452600 T^{3} + 6891960488 T^{4} + 34452600 p^{3} T^{5} + 144563 p^{6} T^{6} + 520 p^{9} T^{7} + p^{12} T^{8}$$
37$D_4\times C_2$ $$1 + 7 T - 74033 T^{2} - 190568 T^{3} + 2919934318 T^{4} - 190568 p^{3} T^{5} - 74033 p^{6} T^{6} + 7 p^{9} T^{7} + p^{12} T^{8}$$
41$D_{4}$ $$( 1 + 426 T + 171106 T^{2} + 426 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
43$D_{4}$ $$( 1 + 107 T + 86220 T^{2} + 107 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 + 576 T + 89606 T^{2} + 19885824 T^{3} + 13421113923 T^{4} + 19885824 p^{3} T^{5} + 89606 p^{6} T^{6} + 576 p^{9} T^{7} + p^{12} T^{8}$$
53$D_4\times C_2$ $$1 + 243 T - 250441 T^{2} + 2851848 T^{3} + 64828660998 T^{4} + 2851848 p^{3} T^{5} - 250441 p^{6} T^{6} + 243 p^{9} T^{7} + p^{12} T^{8}$$
59$D_4\times C_2$ $$1 - 7 T - 200565 T^{2} + 1471008 T^{3} - 1944620216 T^{4} + 1471008 p^{3} T^{5} - 200565 p^{6} T^{6} - 7 p^{9} T^{7} + p^{12} T^{8}$$
61$D_4\times C_2$ $$1 - 224 T - 410950 T^{2} - 1604736 T^{3} + 149727814859 T^{4} - 1604736 p^{3} T^{5} - 410950 p^{6} T^{6} - 224 p^{9} T^{7} + p^{12} T^{8}$$
67$D_4\times C_2$ $$1 + 687 T - 206863 T^{2} + 53109222 T^{3} + 228403689708 T^{4} + 53109222 p^{3} T^{5} - 206863 p^{6} T^{6} + 687 p^{9} T^{7} + p^{12} T^{8}$$
71$D_{4}$ $$( 1 + 472 T + 637018 T^{2} + 472 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
73$D_4\times C_2$ $$1 + 921 T + 270053 T^{2} - 184058166 T^{3} - 147013032042 T^{4} - 184058166 p^{3} T^{5} + 270053 p^{6} T^{6} + 921 p^{9} T^{7} + p^{12} T^{8}$$
79$D_4\times C_2$ $$1 - 526 T - 757051 T^{2} - 25063374 T^{3} + 689091996644 T^{4} - 25063374 p^{3} T^{5} - 757051 p^{6} T^{6} - 526 p^{9} T^{7} + p^{12} T^{8}$$
83$D_{4}$ $$( 1 + 221 T + 945628 T^{2} + 221 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
89$D_4\times C_2$ $$1 + 774 T - 367486 T^{2} - 343173024 T^{3} + 14930800239 T^{4} - 343173024 p^{3} T^{5} - 367486 p^{6} T^{6} + 774 p^{9} T^{7} + p^{12} T^{8}$$
97$D_{4}$ $$( 1 + 1953 T + 2366992 T^{2} + 1953 p^{3} T^{3} + p^{6} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$