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

Origins of factors

Downloads

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Normalization:  

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

Graph of the $Z$-function along the critical line