# Properties

 Label 8-136e4-1.1-c1e4-0-0 Degree $8$ Conductor $342102016$ Sign $1$ Analytic cond. $1.39079$ Root an. cond. $1.04209$ 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 + 4·5-s − 4·7-s + 10·9-s + 4·11-s − 16·15-s + 4·17-s − 12·19-s + 16·21-s + 4·23-s + 10·25-s − 20·27-s − 12·29-s + 12·31-s − 16·33-s − 16·35-s + 4·37-s − 12·41-s + 12·43-s + 40·45-s − 6·49-s − 16·51-s + 20·53-s + 16·55-s + 48·57-s + 28·59-s − 28·61-s + ⋯
 L(s)  = 1 − 2.30·3-s + 1.78·5-s − 1.51·7-s + 10/3·9-s + 1.20·11-s − 4.13·15-s + 0.970·17-s − 2.75·19-s + 3.49·21-s + 0.834·23-s + 2·25-s − 3.84·27-s − 2.22·29-s + 2.15·31-s − 2.78·33-s − 2.70·35-s + 0.657·37-s − 1.87·41-s + 1.82·43-s + 5.96·45-s − 6/7·49-s − 2.24·51-s + 2.74·53-s + 2.15·55-s + 6.35·57-s + 3.64·59-s − 3.58·61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{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 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{12} \cdot 17^{4}$$ Sign: $1$ Analytic conductor: $$1.39079$$ Root analytic conductor: $$1.04209$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{136} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{12} \cdot 17^{4} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.6597216632$$ $$L(\frac12)$$ $$\approx$$ $$0.6597216632$$ $$L(\frac{3}{2})$$ 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 $$1$$
17$C_2$ $$( 1 - 2 T + p T^{2} )^{2}$$
good3$C_2^2$$\times$$C_2^2$ $$( 1 - 2 T^{2} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )$$
5$D_4\times C_2$ $$1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
7$D_4\times C_2$ $$1 + 4 T + 22 T^{2} + 68 T^{3} + 226 T^{4} + 68 p T^{5} + 22 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
11$D_4\times C_2$ $$1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
13$C_2$ $$( 1 - p T^{2} )^{4}$$
19$D_4\times C_2$ $$1 + 12 T + 72 T^{2} + 396 T^{3} + 1982 T^{4} + 396 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
23$D_4\times C_2$ $$1 - 4 T + 22 T^{2} + 4 p T^{3} - 18 p T^{4} + 4 p^{2} T^{5} + 22 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
29$D_4\times C_2$ $$1 + 12 T + 54 T^{2} + 108 T^{3} + 162 T^{4} + 108 p T^{5} + 54 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
31$D_4\times C_2$ $$1 - 12 T + 54 T^{2} - 108 T^{3} + 162 T^{4} - 108 p T^{5} + 54 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
37$D_4\times C_2$ $$1 - 4 T + 6 T^{2} - 4 T^{3} + 2 T^{4} - 4 p T^{5} + 6 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
41$D_4\times C_2$ $$1 + 12 T + 134 T^{2} + 1132 T^{3} + 8450 T^{4} + 1132 p T^{5} + 134 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
43$D_4\times C_2$ $$1 - 12 T + 72 T^{2} - 300 T^{3} + 926 T^{4} - 300 p T^{5} + 72 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
47$C_4\times C_2$ $$1 + 28 T^{2} + 1414 T^{4} + 28 p^{2} T^{6} + p^{4} T^{8}$$
53$D_4\times C_2$ $$1 - 20 T + 200 T^{2} - 1740 T^{3} + 13982 T^{4} - 1740 p T^{5} + 200 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8}$$
59$D_4\times C_2$ $$1 - 28 T + 392 T^{2} - 4284 T^{3} + 37982 T^{4} - 4284 p T^{5} + 392 p^{2} T^{6} - 28 p^{3} T^{7} + p^{4} T^{8}$$
61$D_4\times C_2$ $$1 + 28 T + 438 T^{2} + 4892 T^{3} + 43170 T^{4} + 4892 p T^{5} + 438 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8}$$
67$D_{4}$ $$( 1 - 8 T + 118 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2}$$
71$D_4\times C_2$ $$1 - 12 T + 134 T^{2} - 1612 T^{3} + 14210 T^{4} - 1612 p T^{5} + 134 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8}$$
73$D_4\times C_2$ $$1 - 4 T + 102 T^{2} + 124 T^{3} + 3138 T^{4} + 124 p T^{5} + 102 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
79$D_4\times C_2$ $$1 - 36 T + 662 T^{2} - 8388 T^{3} + 83042 T^{4} - 8388 p T^{5} + 662 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8}$$
83$D_4\times C_2$ $$1 + 12 T + 72 T^{2} + 780 T^{3} + 8126 T^{4} + 780 p T^{5} + 72 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8}$$
89$D_4\times C_2$ $$1 - 260 T^{2} + 30694 T^{4} - 260 p^{2} T^{6} + p^{4} T^{8}$$
97$D_4\times C_2$ $$1 + 28 T + 214 T^{2} - 1828 T^{3} - 42398 T^{4} - 1828 p T^{5} + 214 p^{2} T^{6} + 28 p^{3} T^{7} + p^{4} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$