# Properties

 Label 8-6e4-1.1-c10e4-0-0 Degree $8$ Conductor $1296$ Sign $1$ Analytic cond. $211.191$ Root an. cond. $1.95247$ Motivic weight $10$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 + 84·3-s − 1.02e3·4-s − 4.51e4·7-s + 8.30e4·9-s − 8.60e4·12-s + 2.75e5·13-s + 7.86e5·16-s − 1.56e6·19-s − 3.78e6·21-s + 2.66e6·25-s + 1.83e7·27-s + 4.61e7·28-s − 2.17e7·31-s − 8.50e7·36-s − 7.10e7·37-s + 2.31e7·39-s − 4.70e8·43-s + 6.60e7·48-s + 4.27e8·49-s − 2.81e8·52-s − 1.31e8·57-s − 1.18e9·61-s − 3.74e9·63-s − 5.36e8·64-s − 2.97e8·67-s + 6.53e9·73-s + 2.23e8·75-s + ⋯
 L(s)  = 1 + 0.345·3-s − 4-s − 2.68·7-s + 1.40·9-s − 0.345·12-s + 0.741·13-s + 3/4·16-s − 0.633·19-s − 0.927·21-s + 0.272·25-s + 1.27·27-s + 2.68·28-s − 0.760·31-s − 1.40·36-s − 1.02·37-s + 0.256·39-s − 3.20·43-s + 7/27·48-s + 1.51·49-s − 0.741·52-s − 0.219·57-s − 1.40·61-s − 3.77·63-s − 1/2·64-s − 0.220·67-s + 3.15·73-s + 0.0943·75-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+5)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$1296$$    =    $$2^{4} \cdot 3^{4}$$ Sign: $1$ Analytic conductor: $$211.191$$ Root analytic conductor: $$1.95247$$ Motivic weight: $$10$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{6} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 1296,\ (\ :5, 5, 5, 5),\ 1)$$

## Particular Values

 $$L(\frac{11}{2})$$ $$\approx$$ $$0.6382517833$$ $$L(\frac12)$$ $$\approx$$ $$0.6382517833$$ $$L(6)$$ 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$C_2$ $$( 1 + p^{9} T^{2} )^{2}$$
3$C_2^2$ $$1 - 28 p T - 938 p^{4} T^{2} - 28 p^{11} T^{3} + p^{20} T^{4}$$
good5$D_4\times C_2$ $$1 - 533012 p T^{2} + 6693053487126 p^{2} T^{4} - 533012 p^{21} T^{6} + p^{40} T^{8}$$
7$D_{4}$ $$( 1 + 22556 T + 78482346 p T^{2} + 22556 p^{10} T^{3} + p^{20} T^{4} )^{2}$$
11$D_4\times C_2$ $$1 - 4130589740 p T^{2} +$$$$12\!\cdots\!02$$$$T^{4} - 4130589740 p^{21} T^{6} + p^{40} T^{8}$$
13$D_{4}$ $$( 1 - 137620 T + 223344855798 T^{2} - 137620 p^{10} T^{3} + p^{20} T^{4} )^{2}$$
17$D_4\times C_2$ $$1 - 503008419460 T^{2} -$$$$28\!\cdots\!38$$$$T^{4} - 503008419460 p^{20} T^{6} + p^{40} T^{8}$$
19$D_{4}$ $$( 1 + 784364 T + 11072641146486 T^{2} + 784364 p^{10} T^{3} + p^{20} T^{4} )^{2}$$
23$D_4\times C_2$ $$1 - 132650537376580 T^{2} +$$$$14\!\cdots\!78$$$$p^{2} T^{4} - 132650537376580 p^{20} T^{6} + p^{40} T^{8}$$
29$D_4\times C_2$ $$1 - 775524833463844 T^{2} +$$$$31\!\cdots\!86$$$$T^{4} - 775524833463844 p^{20} T^{6} + p^{40} T^{8}$$
31$D_{4}$ $$( 1 + 10892924 T + 345177991175046 T^{2} + 10892924 p^{10} T^{3} + p^{20} T^{4} )^{2}$$
37$D_{4}$ $$( 1 + 35507084 T + 5319849690001302 T^{2} + 35507084 p^{10} T^{3} + p^{20} T^{4} )^{2}$$
41$D_4\times C_2$ $$1 - 30129232679351620 T^{2} +$$$$47\!\cdots\!42$$$$T^{4} - 30129232679351620 p^{20} T^{6} + p^{40} T^{8}$$
43$D_{4}$ $$( 1 + 5473124 p T + 50501107105165014 T^{2} + 5473124 p^{11} T^{3} + p^{20} T^{4} )^{2}$$
47$D_4\times C_2$ $$1 - 185271765343089796 T^{2} +$$$$13\!\cdots\!06$$$$T^{4} - 185271765343089796 p^{20} T^{6} + p^{40} T^{8}$$
53$D_4\times C_2$ $$1 - 486913415004520420 T^{2} +$$$$10\!\cdots\!62$$$$T^{4} - 486913415004520420 p^{20} T^{6} + p^{40} T^{8}$$
59$D_4\times C_2$ $$1 - 1720402455795118180 T^{2} +$$$$12\!\cdots\!62$$$$T^{4} - 1720402455795118180 p^{20} T^{6} + p^{40} T^{8}$$
61$D_{4}$ $$( 1 + 592019372 T + 1459215526973846838 T^{2} + 592019372 p^{10} T^{3} + p^{20} T^{4} )^{2}$$
67$D_{4}$ $$( 1 + 148682924 T + 3295467847639477302 T^{2} + 148682924 p^{10} T^{3} + p^{20} T^{4} )^{2}$$
71$D_4\times C_2$ $$1 - 11173485987747195844 T^{2} +$$$$52\!\cdots\!86$$$$T^{4} - 11173485987747195844 p^{20} T^{6} + p^{40} T^{8}$$
73$D_{4}$ $$( 1 - 3267134500 T + 10958748572669742438 T^{2} - 3267134500 p^{10} T^{3} + p^{20} T^{4} )^{2}$$
79$D_{4}$ $$( 1 - 99641284 T + 15267587176773126726 T^{2} - 99641284 p^{10} T^{3} + p^{20} T^{4} )^{2}$$
83$D_4\times C_2$ $$1 - 56087025146752445092 T^{2} +$$$$12\!\cdots\!58$$$$T^{4} - 56087025146752445092 p^{20} T^{6} + p^{40} T^{8}$$
89$D_4\times C_2$ $$1 - 97642754408129363140 T^{2} +$$$$41\!\cdots\!62$$$$T^{4} - 97642754408129363140 p^{20} T^{6} + p^{40} T^{8}$$
97$D_{4}$ $$( 1 + 19588177532 T +$$$$24\!\cdots\!94$$$$T^{2} + 19588177532 p^{10} T^{3} + p^{20} T^{4} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$