# Properties

 Label 8-11e8-1.1-c3e4-0-0 Degree $8$ Conductor $214358881$ Sign $1$ Analytic cond. $2597.80$ Root an. cond. $2.67193$ Motivic weight $3$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 8·3-s + 8·4-s − 18·5-s + 27·9-s − 64·12-s + 144·15-s − 144·20-s − 432·23-s + 125·25-s − 340·31-s + 216·36-s + 434·37-s − 486·45-s + 36·47-s + 343·49-s + 738·53-s + 720·59-s + 1.15e3·60-s − 1.66e3·67-s + 3.45e3·69-s − 612·71-s − 1.00e3·75-s + 6.69e3·89-s − 3.45e3·92-s + 2.72e3·93-s + 34·97-s + 1.00e3·100-s + ⋯
 L(s)  = 1 − 1.53·3-s + 4-s − 1.60·5-s + 9-s − 1.53·12-s + 2.47·15-s − 1.60·20-s − 3.91·23-s + 25-s − 1.96·31-s + 36-s + 1.92·37-s − 1.60·45-s + 0.111·47-s + 49-s + 1.91·53-s + 1.58·59-s + 2.47·60-s − 3.03·67-s + 6.02·69-s − 1.02·71-s − 1.53·75-s + 7.97·89-s − 3.91·92-s + 3.03·93-s + 0.0355·97-s + 100-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut &\left(11^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

## Invariants

 Degree: $$8$$ Conductor: $$11^{8}$$ Sign: $1$ Analytic conductor: $$2597.80$$ Root analytic conductor: $$2.67193$$ Motivic weight: $$3$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 11^{8} ,\ ( \ : 3/2, 3/2, 3/2, 3/2 ),\ 1 )$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$0.5995704265$$ $$L(\frac12)$$ $$\approx$$ $$0.5995704265$$ $$L(\frac{5}{2})$$ 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)$
bad11 $$1$$
good2$C_4\times C_2$ $$1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8}$$
3$C_4\times C_2$ $$1 + 8 T + 37 T^{2} + 80 T^{3} - 359 T^{4} + 80 p^{3} T^{5} + 37 p^{6} T^{6} + 8 p^{9} T^{7} + p^{12} T^{8}$$
5$C_4\times C_2$ $$1 + 18 T + 199 T^{2} + 1332 T^{3} - 899 T^{4} + 1332 p^{3} T^{5} + 199 p^{6} T^{6} + 18 p^{9} T^{7} + p^{12} T^{8}$$
7$C_4\times C_2$ $$1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8}$$
13$C_4\times C_2$ $$1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8}$$
17$C_4\times C_2$ $$1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8}$$
19$C_4\times C_2$ $$1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8}$$
23$C_2$ $$( 1 + 108 T + p^{3} T^{2} )^{4}$$
29$C_4\times C_2$ $$1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8}$$
31$C_4\times C_2$ $$1 + 340 T + 85809 T^{2} + 19046120 T^{3} + 3919344881 T^{4} + 19046120 p^{3} T^{5} + 85809 p^{6} T^{6} + 340 p^{9} T^{7} + p^{12} T^{8}$$
37$C_4\times C_2$ $$1 - 434 T + 137703 T^{2} - 37779700 T^{3} + 9421319741 T^{4} - 37779700 p^{3} T^{5} + 137703 p^{6} T^{6} - 434 p^{9} T^{7} + p^{12} T^{8}$$
41$C_4\times C_2$ $$1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8}$$
43$C_2$ $$( 1 + p^{3} T^{2} )^{4}$$
47$C_4\times C_2$ $$1 - 36 T - 102527 T^{2} + 7428600 T^{3} + 10377231121 T^{4} + 7428600 p^{3} T^{5} - 102527 p^{6} T^{6} - 36 p^{9} T^{7} + p^{12} T^{8}$$
53$C_4\times C_2$ $$1 - 738 T + 395767 T^{2} - 182204820 T^{3} + 75546553501 T^{4} - 182204820 p^{3} T^{5} + 395767 p^{6} T^{6} - 738 p^{9} T^{7} + p^{12} T^{8}$$
59$C_4\times C_2$ $$1 - 720 T + 313021 T^{2} - 77502240 T^{3} - 8486327159 T^{4} - 77502240 p^{3} T^{5} + 313021 p^{6} T^{6} - 720 p^{9} T^{7} + p^{12} T^{8}$$
61$C_4\times C_2$ $$1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8}$$
67$C_2$ $$( 1 + 416 T + p^{3} T^{2} )^{4}$$
71$C_4\times C_2$ $$1 + 612 T + 16633 T^{2} - 208862136 T^{3} - 133776760895 T^{4} - 208862136 p^{3} T^{5} + 16633 p^{6} T^{6} + 612 p^{9} T^{7} + p^{12} T^{8}$$
73$C_4\times C_2$ $$1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8}$$
79$C_4\times C_2$ $$1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8}$$
83$C_4\times C_2$ $$1 - p^{3} T^{2} + p^{6} T^{4} - p^{9} T^{6} + p^{12} T^{8}$$
89$C_2$ $$( 1 - 1674 T + p^{3} T^{2} )^{4}$$
97$C_4\times C_2$ $$1 - 34 T - 911517 T^{2} + 62022460 T^{3} + 829808191301 T^{4} + 62022460 p^{3} T^{5} - 911517 p^{6} T^{6} - 34 p^{9} T^{7} + p^{12} T^{8}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$