# Properties

 Label 8-363e4-1.1-c1e4-0-1 Degree $8$ Conductor $17363069361$ Sign $1$ Analytic cond. $70.5886$ Root an. cond. $1.70251$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·2-s + 3-s + 2·4-s − 4·5-s − 2·6-s + 7-s + 8·10-s + 2·12-s − 2·13-s − 2·14-s − 4·15-s + 4·17-s − 3·19-s − 8·20-s + 21-s + 8·23-s + 5·25-s + 4·26-s + 2·28-s + 6·29-s + 8·30-s + 5·31-s − 8·32-s − 8·34-s − 4·35-s − 3·37-s + 6·38-s + ⋯
 L(s)  = 1 − 1.41·2-s + 0.577·3-s + 4-s − 1.78·5-s − 0.816·6-s + 0.377·7-s + 2.52·10-s + 0.577·12-s − 0.554·13-s − 0.534·14-s − 1.03·15-s + 0.970·17-s − 0.688·19-s − 1.78·20-s + 0.218·21-s + 1.66·23-s + 25-s + 0.784·26-s + 0.377·28-s + 1.11·29-s + 1.46·30-s + 0.898·31-s − 1.41·32-s − 1.37·34-s − 0.676·35-s − 0.493·37-s + 0.973·38-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$8$$ Conductor: $$3^{4} \cdot 11^{8}$$ Sign: $1$ Analytic conductor: $$70.5886$$ Root analytic conductor: $$1.70251$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: Trivial Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(8,\ 3^{4} \cdot 11^{8} ,\ ( \ : 1/2, 1/2, 1/2, 1/2 ),\ 1 )$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.4651487290$$ $$L(\frac12)$$ $$\approx$$ $$0.4651487290$$ $$L(\frac{3}{2})$$ 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)$
bad3$C_4$ $$1 - T + T^{2} - T^{3} + T^{4}$$
11 $$1$$
good2$C_4\times C_2$ $$1 + p T + p T^{2} - p^{2} T^{4} + p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8}$$
5$C_4\times C_2$ $$1 + 4 T + 11 T^{2} + 24 T^{3} + 41 T^{4} + 24 p T^{5} + 11 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8}$$
7$C_4\times C_2$ $$1 - T - 6 T^{2} + 13 T^{3} + 29 T^{4} + 13 p T^{5} - 6 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8}$$
13$C_4\times C_2$ $$1 + 2 T - 9 T^{2} - 44 T^{3} + 29 T^{4} - 44 p T^{5} - 9 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
17$C_4\times C_2$ $$1 - 4 T - T^{2} + 72 T^{3} - 271 T^{4} + 72 p T^{5} - p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8}$$
19$C_4\times C_2$ $$1 + 3 T - 10 T^{2} - 87 T^{3} - 71 T^{4} - 87 p T^{5} - 10 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
23$C_2$ $$( 1 - 2 T + p T^{2} )^{4}$$
29$C_4\times C_2$ $$1 - 6 T + 7 T^{2} + 132 T^{3} - 995 T^{4} + 132 p T^{5} + 7 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
31$C_4\times C_2$ $$1 - 5 T - 6 T^{2} + 185 T^{3} - 739 T^{4} + 185 p T^{5} - 6 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8}$$
37$C_4\times C_2$ $$1 + 3 T - 28 T^{2} - 195 T^{3} + 451 T^{4} - 195 p T^{5} - 28 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8}$$
41$C_4\times C_2$ $$1 + 2 T - 37 T^{2} - 156 T^{3} + 1205 T^{4} - 156 p T^{5} - 37 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
43$C_2$ $$( 1 + 12 T + p T^{2} )^{4}$$
47$C_4\times C_2$ $$1 + 2 T - 43 T^{2} - 180 T^{3} + 1661 T^{4} - 180 p T^{5} - 43 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8}$$
53$C_4\times C_2$ $$1 + 6 T - 17 T^{2} - 420 T^{3} - 1619 T^{4} - 420 p T^{5} - 17 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8}$$
59$C_4\times C_2$ $$1 - 10 T + 41 T^{2} + 180 T^{3} - 4219 T^{4} + 180 p T^{5} + 41 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8}$$
61$C_4\times C_2$ $$1 - 3 T - 52 T^{2} + 339 T^{3} + 2155 T^{4} + 339 p T^{5} - 52 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8}$$
67$C_2$ $$( 1 + T + p T^{2} )^{4}$$
71$C_4\times C_2$ $$1 - p T^{2} + p^{2} T^{4} - p^{3} T^{6} + p^{4} T^{8}$$
73$C_4\times C_2$ $$1 + 11 T + 48 T^{2} - 275 T^{3} - 6529 T^{4} - 275 p T^{5} + 48 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8}$$
79$C_4\times C_2$ $$1 - 11 T + 42 T^{2} + 407 T^{3} - 7795 T^{4} + 407 p T^{5} + 42 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8}$$
83$C_4\times C_2$ $$1 - 6 T - 47 T^{2} + 780 T^{3} - 779 T^{4} + 780 p T^{5} - 47 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8}$$
89$C_2$ $$( 1 - 12 T + p T^{2} )^{4}$$
97$C_4\times C_2$ $$1 + 5 T - 72 T^{2} - 845 T^{3} + 2759 T^{4} - 845 p T^{5} - 72 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−8.414590856015217969918428000317, −8.095960379650779406713820007232, −7.88737758257977206495171249189, −7.68927421985911301835714850372, −7.42244813729472633909546977854, −7.28162140044125638184862420894, −6.77916747609704950104615633233, −6.72051016256054584965644855639, −6.56529712998703429070053798983, −6.23154485786090029282553865122, −5.72102850617594003816902025012, −5.10108224300748058215309484366, −5.02197415486427816098734713618, −5.00891944400626884196175299346, −4.76120497561502996147941142752, −4.19009347778286151039189145019, −3.61281034113458789604522528445, −3.59716559281281209627703992637, −3.48844896291684355147802204782, −2.82630080265690501179832189184, −2.79451118025348070337925609331, −1.83813456029861339633822950396, −1.77342867087038813933050586413, −1.15785176556516755631409637777, −0.38443575449745238907659246676, 0.38443575449745238907659246676, 1.15785176556516755631409637777, 1.77342867087038813933050586413, 1.83813456029861339633822950396, 2.79451118025348070337925609331, 2.82630080265690501179832189184, 3.48844896291684355147802204782, 3.59716559281281209627703992637, 3.61281034113458789604522528445, 4.19009347778286151039189145019, 4.76120497561502996147941142752, 5.00891944400626884196175299346, 5.02197415486427816098734713618, 5.10108224300748058215309484366, 5.72102850617594003816902025012, 6.23154485786090029282553865122, 6.56529712998703429070053798983, 6.72051016256054584965644855639, 6.77916747609704950104615633233, 7.28162140044125638184862420894, 7.42244813729472633909546977854, 7.68927421985911301835714850372, 7.88737758257977206495171249189, 8.095960379650779406713820007232, 8.414590856015217969918428000317