# Properties

 Label 8-912e4-1.1-c1e4-0-2 Degree $8$ Conductor $691798081536$ Sign $1$ Analytic cond. $2812.46$ Root an. cond. $2.69858$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 3-s − 3·5-s − 2·7-s + 3·9-s + 3·15-s − 3·17-s + 16·19-s + 2·21-s − 3·23-s − 2·25-s − 8·27-s − 5·29-s + 6·35-s − 7·41-s + 12·43-s − 9·45-s + 9·47-s − 9·49-s + 3·51-s + 17·53-s − 16·57-s + 9·59-s + 2·61-s − 6·63-s − 18·67-s + 3·69-s + 19·71-s + ⋯
 L(s)  = 1 − 0.577·3-s − 1.34·5-s − 0.755·7-s + 9-s + 0.774·15-s − 0.727·17-s + 3.67·19-s + 0.436·21-s − 0.625·23-s − 2/5·25-s − 1.53·27-s − 0.928·29-s + 1.01·35-s − 1.09·41-s + 1.82·43-s − 1.34·45-s + 1.31·47-s − 9/7·49-s + 0.420·51-s + 2.33·53-s − 2.11·57-s + 1.17·59-s + 0.256·61-s − 0.755·63-s − 2.19·67-s + 0.361·69-s + 2.25·71-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 19^{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 3^{4} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{16} \cdot 3^{4} \cdot 19^{4}$$ Sign: $1$ Analytic conductor: $$2812.46$$ Root analytic conductor: $$2.69858$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{912} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{16} \cdot 3^{4} \cdot 19^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.6726047614$$ $$L(\frac12)$$ $$\approx$$ $$0.6726047614$$ $$L(\frac{3}{2})$$ 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$C_2^2$ $$1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4}$$
19$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
good5$D_4\times C_2$ $$1 + 3 T + 11 T^{2} + 24 T^{3} + 54 T^{4} + 24 p T^{5} + 11 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
7$D_{4}$ $$( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} )^{2}$$
11$C_2^2$ $$( 1 - 10 T^{2} + p^{2} T^{4} )^{2}$$
13$C_2^3$ $$1 + 15 T^{2} + 56 T^{4} + 15 p^{2} T^{6} + p^{4} T^{8}$$
17$D_4\times C_2$ $$1 + 3 T + 35 T^{2} + 96 T^{3} + 786 T^{4} + 96 p T^{5} + 35 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
23$D_4\times C_2$ $$1 + 3 T - 19 T^{2} - 66 T^{3} + 24 T^{4} - 66 p T^{5} - 19 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
29$D_4\times C_2$ $$1 + 5 T - 31 T^{2} - 10 T^{3} + 1570 T^{4} - 10 p T^{5} - 31 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
31$D_4\times C_2$ $$1 - 45 T^{2} + 2024 T^{4} - 45 p^{2} T^{6} + p^{4} T^{8}$$
37$D_4\times C_2$ $$1 + 39 T^{2} + 2120 T^{4} + 39 p^{2} T^{6} + p^{4} T^{8}$$
41$D_4\times C_2$ $$1 + 7 T - 37 T^{2} + 28 T^{3} + 3706 T^{4} + 28 p T^{5} - 37 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8}$$
43$D_4\times C_2$ $$1 - 12 T + 55 T^{2} - 36 T^{3} - 120 T^{4} - 36 p T^{5} + 55 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
47$D_4\times C_2$ $$1 - 9 T + 125 T^{2} - 882 T^{3} + 8664 T^{4} - 882 p T^{5} + 125 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}$$
53$D_4\times C_2$ $$1 - 17 T + 119 T^{2} - 1088 T^{3} + 10774 T^{4} - 1088 p T^{5} + 119 p^{2} T^{6} - 17 p^{3} T^{7} + p^{4} T^{8}$$
59$D_4\times C_2$ $$1 - 9 T - 49 T^{2} - 108 T^{3} + 8640 T^{4} - 108 p T^{5} - 49 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}$$
61$C_2$ $$( 1 - 14 T + p T^{2} )^{2}( 1 + 13 T + p T^{2} )^{2}$$
67$D_4\times C_2$ $$1 + 18 T + 225 T^{2} + 2106 T^{3} + 16436 T^{4} + 2106 p T^{5} + 225 p^{2} T^{6} + 18 p^{3} T^{7} + p^{4} T^{8}$$
71$D_4\times C_2$ $$1 - 19 T + 137 T^{2} - 1558 T^{3} + 19504 T^{4} - 1558 p T^{5} + 137 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8}$$
73$C_2^3$ $$1 - 113 T^{2} + 7440 T^{4} - 113 p^{2} T^{6} + p^{4} T^{8}$$
79$D_4\times C_2$ $$1 - 6 T - 3 T^{2} + 90 T^{3} - 5068 T^{4} + 90 p T^{5} - 3 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
83$C_2^2$ $$( 1 - 154 T^{2} + p^{2} T^{4} )^{2}$$
89$D_4\times C_2$ $$1 - 9 T - 109 T^{2} - 108 T^{3} + 20970 T^{4} - 108 p T^{5} - 109 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8}$$
97$D_4\times C_2$ $$1 - 27 T + 473 T^{2} - 6210 T^{3} + 67062 T^{4} - 6210 p T^{5} + 473 p^{2} T^{6} - 27 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$