Properties

Label 8-21e8-1.1-c5e4-0-1
Degree $8$
Conductor $37822859361$
Sign $1$
Analytic cond. $2.50262\times 10^{7}$
Root an. cond. $8.41006$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 25·4-s + 75·8-s − 402·11-s + 462·13-s + 599·16-s − 276·17-s + 510·19-s + 1.20e3·22-s − 6.90e3·23-s − 4.84e3·25-s − 1.38e3·26-s − 540·29-s − 6.41e3·31-s − 4.06e3·32-s + 828·34-s + 1.52e4·37-s − 1.53e3·38-s − 4.30e3·41-s + 2.91e4·43-s + 1.00e4·44-s + 2.07e4·46-s + 1.50e4·47-s + 1.45e4·50-s − 1.15e4·52-s − 1.36e4·53-s + 1.62e3·58-s + ⋯
L(s)  = 1  − 0.530·2-s − 0.781·4-s + 0.414·8-s − 1.00·11-s + 0.758·13-s + 0.584·16-s − 0.231·17-s + 0.324·19-s + 0.531·22-s − 2.71·23-s − 1.54·25-s − 0.402·26-s − 0.119·29-s − 1.19·31-s − 0.701·32-s + 0.122·34-s + 1.83·37-s − 0.171·38-s − 0.400·41-s + 2.40·43-s + 0.782·44-s + 1.44·46-s + 0.994·47-s + 0.821·50-s − 0.592·52-s − 0.669·53-s + 0.0632·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]

Invariants

Degree: \(8\)
Conductor: \(3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(2.50262\times 10^{7}\)
Root analytic conductor: \(8.41006\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{441} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((8,\ 3^{8} \cdot 7^{8} ,\ ( \ : 5/2, 5/2, 5/2, 5/2 ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(0.4426908911\)
\(L(\frac12)\) \(\approx\) \(0.4426908911\)
\(L(\frac{7}{2})\) 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 \( 1 \)
7 \( 1 \)
good2$C_2 \wr S_4$ \( 1 + 3 T + 17 p T^{2} + 51 p T^{3} + 83 p^{2} T^{4} + 51 p^{6} T^{5} + 17 p^{11} T^{6} + 3 p^{15} T^{7} + p^{20} T^{8} \)
5$C_2 \wr S_4$ \( 1 + 4843 T^{2} - 12108 p^{2} T^{3} + 7549256 T^{4} - 12108 p^{7} T^{5} + 4843 p^{10} T^{6} + p^{20} T^{8} \)
11$C_2 \wr S_4$ \( 1 + 402 T + 405841 T^{2} + 93416934 T^{3} + 77165216456 T^{4} + 93416934 p^{5} T^{5} + 405841 p^{10} T^{6} + 402 p^{15} T^{7} + p^{20} T^{8} \)
13$C_2 \wr S_4$ \( 1 - 462 T + 336749 T^{2} + 500754 T^{3} + 123849672672 T^{4} + 500754 p^{5} T^{5} + 336749 p^{10} T^{6} - 462 p^{15} T^{7} + p^{20} T^{8} \)
17$C_2 \wr S_4$ \( 1 + 276 T + 4084676 T^{2} + 280763388 T^{3} + 7517219673798 T^{4} + 280763388 p^{5} T^{5} + 4084676 p^{10} T^{6} + 276 p^{15} T^{7} + p^{20} T^{8} \)
19$C_2 \wr S_4$ \( 1 - 510 T + 4007525 T^{2} - 2990607690 T^{3} + 14975063190924 T^{4} - 2990607690 p^{5} T^{5} + 4007525 p^{10} T^{6} - 510 p^{15} T^{7} + p^{20} T^{8} \)
23$C_2 \wr S_4$ \( 1 + 300 p T + 32867228 T^{2} + 116109363780 T^{3} + 343244422915686 T^{4} + 116109363780 p^{5} T^{5} + 32867228 p^{10} T^{6} + 300 p^{16} T^{7} + p^{20} T^{8} \)
29$C_2 \wr S_4$ \( 1 + 540 T + 29394199 T^{2} - 29299685892 T^{3} + 772430149366772 T^{4} - 29299685892 p^{5} T^{5} + 29394199 p^{10} T^{6} + 540 p^{15} T^{7} + p^{20} T^{8} \)
31$C_2 \wr S_4$ \( 1 + 6410 T + 96802736 T^{2} + 491920044696 T^{3} + 3990219807562217 T^{4} + 491920044696 p^{5} T^{5} + 96802736 p^{10} T^{6} + 6410 p^{15} T^{7} + p^{20} T^{8} \)
37$C_2 \wr S_4$ \( 1 - 15250 T + 230519621 T^{2} - 1675418532570 T^{3} + 17310190338885440 T^{4} - 1675418532570 p^{5} T^{5} + 230519621 p^{10} T^{6} - 15250 p^{15} T^{7} + p^{20} T^{8} \)
41$C_2 \wr S_4$ \( 1 + 4308 T + 270683560 T^{2} + 2598901038204 T^{3} + 34018560333944366 T^{4} + 2598901038204 p^{5} T^{5} + 270683560 p^{10} T^{6} + 4308 p^{15} T^{7} + p^{20} T^{8} \)
43$C_2 \wr S_4$ \( 1 - 29198 T + 787995265 T^{2} - 13076978932730 T^{3} + 187469286625894360 T^{4} - 13076978932730 p^{5} T^{5} + 787995265 p^{10} T^{6} - 29198 p^{15} T^{7} + p^{20} T^{8} \)
47$C_2 \wr S_4$ \( 1 - 15060 T + 968029772 T^{2} - 10316869428852 T^{3} + 338556639544077654 T^{4} - 10316869428852 p^{5} T^{5} + 968029772 p^{10} T^{6} - 15060 p^{15} T^{7} + p^{20} T^{8} \)
53$C_2 \wr S_4$ \( 1 + 13692 T + 1174874543 T^{2} + 9138918677148 T^{3} + 624374053991775828 T^{4} + 9138918677148 p^{5} T^{5} + 1174874543 p^{10} T^{6} + 13692 p^{15} T^{7} + p^{20} T^{8} \)
59$C_2 \wr S_4$ \( 1 + 34830 T + 3047953637 T^{2} + 74240819598150 T^{3} + 3333299875817266188 T^{4} + 74240819598150 p^{5} T^{5} + 3047953637 p^{10} T^{6} + 34830 p^{15} T^{7} + p^{20} T^{8} \)
61$C_2 \wr S_4$ \( 1 + 5364 T + 2845789028 T^{2} + 10211521644252 T^{3} + 3398342140250230278 T^{4} + 10211521644252 p^{5} T^{5} + 2845789028 p^{10} T^{6} + 5364 p^{15} T^{7} + p^{20} T^{8} \)
67$C_2 \wr S_4$ \( 1 + 5994 T + 2554505537 T^{2} + 29219705413074 T^{3} + 3802215802346578704 T^{4} + 29219705413074 p^{5} T^{5} + 2554505537 p^{10} T^{6} + 5994 p^{15} T^{7} + p^{20} T^{8} \)
71$C_2 \wr S_4$ \( 1 + 89268 T + 8738662172 T^{2} + 466802240277492 T^{3} + 25044546792022133910 T^{4} + 466802240277492 p^{5} T^{5} + 8738662172 p^{10} T^{6} + 89268 p^{15} T^{7} + p^{20} T^{8} \)
73$C_2 \wr S_4$ \( 1 - 59638 T + 8655226553 T^{2} - 350574157952982 T^{3} + 27168426643986231860 T^{4} - 350574157952982 p^{5} T^{5} + 8655226553 p^{10} T^{6} - 59638 p^{15} T^{7} + p^{20} T^{8} \)
79$C_2 \wr S_4$ \( 1 + 44062 T + 8526420688 T^{2} + 421600841789848 T^{3} + 33701233575740693881 T^{4} + 421600841789848 p^{5} T^{5} + 8526420688 p^{10} T^{6} + 44062 p^{15} T^{7} + p^{20} T^{8} \)
83$C_2 \wr S_4$ \( 1 - 208446 T + 23363412401 T^{2} - 1724627286611514 T^{3} + \)\(11\!\cdots\!56\)\( T^{4} - 1724627286611514 p^{5} T^{5} + 23363412401 p^{10} T^{6} - 208446 p^{15} T^{7} + p^{20} T^{8} \)
89$C_2 \wr S_4$ \( 1 - 77520 T + 5530024288 T^{2} + 567577230261264 T^{3} - 47925515398937104546 T^{4} + 567577230261264 p^{5} T^{5} + 5530024288 p^{10} T^{6} - 77520 p^{15} T^{7} + p^{20} T^{8} \)
97$C_2 \wr S_4$ \( 1 + 188630 T + 43620869129 T^{2} + 4760435860011510 T^{3} + \)\(59\!\cdots\!64\)\( T^{4} + 4760435860011510 p^{5} T^{5} + 43620869129 p^{10} T^{6} + 188630 p^{15} T^{7} + p^{20} T^{8} \)
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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

−7.40756969004114019803491114645, −7.11078929441611584734525350585, −6.73933817755199891664261293165, −6.26036482269676473445502944229, −6.13951248484428790200474988634, −5.95963113437424457378412958944, −5.61342783483642066867475689257, −5.54945822207082317598282328618, −5.49615048531928508580050510313, −4.67485491593308074048703653781, −4.50726582362199362362575122327, −4.49912709503874631425039267451, −4.11142540964067218662149398226, −3.82535613665763969579904239496, −3.46765948753357563284501076428, −3.40358415375551282883303464034, −2.72838301660620829243733551990, −2.71517759511291102184580802444, −1.98082051413090430184462553238, −1.92398325405100735482365770363, −1.84714680412975526335315305331, −1.01656627101711194647285790961, −0.926751260896687998395792421683, −0.39480709810407028211308373607, −0.14587977994755045465412279064, 0.14587977994755045465412279064, 0.39480709810407028211308373607, 0.926751260896687998395792421683, 1.01656627101711194647285790961, 1.84714680412975526335315305331, 1.92398325405100735482365770363, 1.98082051413090430184462553238, 2.71517759511291102184580802444, 2.72838301660620829243733551990, 3.40358415375551282883303464034, 3.46765948753357563284501076428, 3.82535613665763969579904239496, 4.11142540964067218662149398226, 4.49912709503874631425039267451, 4.50726582362199362362575122327, 4.67485491593308074048703653781, 5.49615048531928508580050510313, 5.54945822207082317598282328618, 5.61342783483642066867475689257, 5.95963113437424457378412958944, 6.13951248484428790200474988634, 6.26036482269676473445502944229, 6.73933817755199891664261293165, 7.11078929441611584734525350585, 7.40756969004114019803491114645

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