# Properties

 Label 8-136e4-1.1-c2e4-0-0 Degree $8$ Conductor $342102016$ Sign $1$ Analytic cond. $188.580$ Root an. cond. $1.92502$ Motivic weight $2$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 8·3-s + 46·9-s − 52·11-s − 16·16-s − 48·17-s + 48·19-s − 184·27-s + 416·33-s − 96·41-s + 28·43-s + 128·48-s + 384·51-s − 384·57-s + 164·59-s + 336·67-s + 48·73-s + 610·81-s + 316·83-s + 428·97-s − 2.39e3·99-s − 524·107-s − 288·113-s + 1.30e3·121-s + 768·123-s + 127-s − 224·129-s + 131-s + ⋯
 L(s)  = 1 − 8/3·3-s + 46/9·9-s − 4.72·11-s − 16-s − 2.82·17-s + 2.52·19-s − 6.81·27-s + 12.6·33-s − 2.34·41-s + 0.651·43-s + 8/3·48-s + 7.52·51-s − 6.73·57-s + 2.77·59-s + 5.01·67-s + 0.657·73-s + 7.53·81-s + 3.80·83-s + 4.41·97-s − 24.1·99-s − 4.89·107-s − 2.54·113-s + 10.7·121-s + 6.24·123-s + 0.00787·127-s − 1.73·129-s + 0.00763·131-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(3-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 17^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$2^{12} \cdot 17^{4}$$ Sign: $1$ Analytic conductor: $$188.580$$ Root analytic conductor: $$1.92502$$ Motivic weight: $$2$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{136} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 2^{12} \cdot 17^{4} ,\ ( \ : 1, 1, 1, 1 ),\ 1 )$$

## Particular Values

 $$L(\frac{3}{2})$$ $$\approx$$ $$0.2240365261$$ $$L(\frac12)$$ $$\approx$$ $$0.2240365261$$ $$L(2)$$ 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$C_2^2$ $$1 + p^{4} T^{4}$$
17$C_2^2$ $$1 + 48 T + 1152 T^{2} + 48 p^{2} T^{3} + p^{4} T^{4}$$
good3$C_2^2$$\times$$C_2^2$ $$( 1 - 14 T^{2} + p^{4} T^{4} )( 1 + 8 T + 32 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )$$
5$C_4\times C_2$ $$1 + p^{8} T^{8}$$
7$C_4\times C_2$ $$1 + p^{8} T^{8}$$
11$C_2$$\times$$C_2^2$ $$( 1 + 14 T + p^{2} T^{2} )^{2}( 1 + 24 T + 288 T^{2} + 24 p^{2} T^{3} + p^{4} T^{4} )$$
13$C_2$ $$( 1 + p^{2} T^{2} )^{4}$$
19$C_2^2$ $$( 1 - 24 T + 288 T^{2} - 24 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
23$C_4\times C_2$ $$1 + p^{8} T^{8}$$
29$C_4\times C_2$ $$1 + p^{8} T^{8}$$
31$C_4\times C_2$ $$1 + p^{8} T^{8}$$
37$C_4\times C_2$ $$1 + p^{8} T^{8}$$
41$C_2^2$$\times$$C_2^2$ $$( 1 - 1246 T^{2} + p^{4} T^{4} )( 1 + 96 T + 4608 T^{2} + 96 p^{2} T^{3} + p^{4} T^{4} )$$
43$C_2$$\times$$C_2^2$ $$( 1 - 14 T + p^{2} T^{2} )^{2}( 1 - 3502 T^{2} + p^{4} T^{4} )$$
47$C_2$ $$( 1 + p^{2} T^{2} )^{4}$$
53$C_2^2$ $$( 1 + p^{4} T^{4} )^{2}$$
59$C_2$$\times$$C_2^2$ $$( 1 - 82 T + p^{2} T^{2} )^{2}( 1 - 238 T^{2} + p^{4} T^{4} )$$
61$C_4\times C_2$ $$1 + p^{8} T^{8}$$
67$C_2^2$ $$( 1 - 168 T + 14112 T^{2} - 168 p^{2} T^{3} + p^{4} T^{4} )^{2}$$
71$C_4\times C_2$ $$1 + p^{8} T^{8}$$
73$C_2^2$$\times$$C_2^2$ $$( 1 - 48 T + 1152 T^{2} - 48 p^{2} T^{3} + p^{4} T^{4} )( 1 + 9506 T^{2} + p^{4} T^{4} )$$
79$C_4\times C_2$ $$1 + p^{8} T^{8}$$
83$C_2$$\times$$C_2^2$ $$( 1 - 158 T + p^{2} T^{2} )^{2}( 1 + 11186 T^{2} + p^{4} T^{4} )$$
89$C_2^2$$\times$$C_2^2$ $$( 1 - 144 T + 10368 T^{2} - 144 p^{2} T^{3} + p^{4} T^{4} )( 1 + 144 T + 10368 T^{2} + 144 p^{2} T^{3} + p^{4} T^{4} )$$
97$C_2$$\times$$C_2^2$ $$( 1 - 94 T + p^{2} T^{2} )^{2}( 1 - 240 T + 28800 T^{2} - 240 p^{2} T^{3} + p^{4} T^{4} )$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$