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

Origins of factors

Downloads

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Normalization:  

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$D_{4}$ \( ( 1 + 19588177532 T + \)\(24\!\cdots\!94\)\( T^{2} + 19588177532 p^{10} T^{3} + p^{20} T^{4} )^{2} \)
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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

−15.45874697311626271191912205813, −14.94666768885243624353242931732, −14.63191173228698903395864232520, −13.68786564713015031078865121610, −13.44648530900008741847180894762, −13.37281674208131396877229763228, −12.73329651514715162216073981973, −12.37259579454103471612235173481, −12.26328058055213093332307633811, −10.96069053696150353265466271595, −10.64154244021490082900169321477, −9.831638671473870957217108698789, −9.681806365547529709097802400295, −9.466190169821971645186608656230, −8.494906272449475173116398799368, −8.314346235040417047998142046242, −7.09930741215260245259768013519, −6.67324175281569157318129879835, −6.30536887493922036140559292646, −5.21672670865504311748330196993, −4.34993008899755257657202656228, −3.44872814696230968742953399596, −3.25733308796666326014871595297, −1.60206598405182890542098816439, −0.32804557147860223561523458314, 0.32804557147860223561523458314, 1.60206598405182890542098816439, 3.25733308796666326014871595297, 3.44872814696230968742953399596, 4.34993008899755257657202656228, 5.21672670865504311748330196993, 6.30536887493922036140559292646, 6.67324175281569157318129879835, 7.09930741215260245259768013519, 8.314346235040417047998142046242, 8.494906272449475173116398799368, 9.466190169821971645186608656230, 9.681806365547529709097802400295, 9.831638671473870957217108698789, 10.64154244021490082900169321477, 10.96069053696150353265466271595, 12.26328058055213093332307633811, 12.37259579454103471612235173481, 12.73329651514715162216073981973, 13.37281674208131396877229763228, 13.44648530900008741847180894762, 13.68786564713015031078865121610, 14.63191173228698903395864232520, 14.94666768885243624353242931732, 15.45874697311626271191912205813

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