Properties

Label 16-21e8-1.1-c5e8-0-0
Degree $16$
Conductor $37822859361$
Sign $1$
Analytic cond. $16559.1$
Root an. cond. $1.83522$
Motivic weight $5$
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  − 3·2-s − 36·3-s + 34·4-s + 108·6-s + 258·7-s − 75·8-s + 486·9-s − 402·11-s − 1.22e3·12-s + 924·13-s − 774·14-s + 251·16-s − 276·17-s − 1.45e3·18-s − 510·19-s − 9.28e3·21-s + 1.20e3·22-s − 6.90e3·23-s + 2.70e3·24-s + 4.84e3·25-s − 2.77e3·26-s + 8.77e3·28-s + 1.08e3·29-s + 6.41e3·31-s − 2.34e3·32-s + 1.44e4·33-s + 828·34-s + ⋯
L(s)  = 1  − 0.530·2-s − 2.30·3-s + 1.06·4-s + 1.22·6-s + 1.99·7-s − 0.414·8-s + 2·9-s − 1.00·11-s − 2.45·12-s + 1.51·13-s − 1.05·14-s + 0.245·16-s − 0.231·17-s − 1.06·18-s − 0.324·19-s − 4.59·21-s + 0.531·22-s − 2.71·23-s + 0.956·24-s + 1.54·25-s − 0.804·26-s + 2.11·28-s + 0.238·29-s + 1.19·31-s − 0.404·32-s + 2.31·33-s + 0.122·34-s + ⋯

Functional equation

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

Invariants

Degree: \(16\)
Conductor: \(3^{8} \cdot 7^{8}\)
Sign: $1$
Analytic conductor: \(16559.1\)
Root analytic conductor: \(1.83522\)
Motivic weight: \(5\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{21} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((16,\ 3^{8} \cdot 7^{8} ,\ ( \ : [5/2]^{8} ),\ 1 )\)

Particular Values

\(L(3)\) \(\approx\) \(0.8054423986\)
\(L(\frac12)\) \(\approx\) \(0.8054423986\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( ( 1 + p^{2} T + p^{4} T^{2} )^{4} \)
7 \( 1 - 258 T + 9344 p T^{2} - 251256 p^{2} T^{3} + 4790655 p^{3} T^{4} - 251256 p^{7} T^{5} + 9344 p^{11} T^{6} - 258 p^{15} T^{7} + p^{20} T^{8} \)
good2 \( 1 + 3 T - 25 T^{2} - 51 p T^{3} + 259 p T^{4} + 1185 p^{2} T^{5} + 14739 p^{2} T^{6} + 1833 p^{3} T^{7} - 116575 p^{4} T^{8} + 1833 p^{8} T^{9} + 14739 p^{12} T^{10} + 1185 p^{17} T^{11} + 259 p^{21} T^{12} - 51 p^{26} T^{13} - 25 p^{30} T^{14} + 3 p^{35} T^{15} + p^{40} T^{16} \)
5 \( 1 - 4843 T^{2} + 24216 p^{2} T^{3} + 15905393 T^{4} - 96476544 p^{2} T^{5} + 149656086942 T^{6} + 275087633352 p^{2} T^{7} - 553760753377714 T^{8} + 275087633352 p^{7} T^{9} + 149656086942 p^{10} T^{10} - 96476544 p^{17} T^{11} + 15905393 p^{20} T^{12} + 24216 p^{27} T^{13} - 4843 p^{30} T^{14} + p^{40} T^{16} \)
11 \( 1 + 402 T - 244237 T^{2} - 23685786 T^{3} + 49988093357 T^{4} - 825774680136 p T^{5} - 7585190415099126 T^{6} + 907765850007682500 T^{7} + \)\(95\!\cdots\!06\)\( T^{8} + 907765850007682500 p^{5} T^{9} - 7585190415099126 p^{10} T^{10} - 825774680136 p^{16} T^{11} + 49988093357 p^{20} T^{12} - 23685786 p^{25} T^{13} - 244237 p^{30} T^{14} + 402 p^{35} T^{15} + p^{40} T^{16} \)
13 \( ( 1 - 462 T + 336749 T^{2} + 500754 T^{3} + 123849672672 T^{4} + 500754 p^{5} T^{5} + 336749 p^{10} T^{6} - 462 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
17 \( 1 + 276 T - 4008500 T^{2} + 565843800 T^{3} + 9089867656090 T^{4} - 2604033925498692 T^{5} - 14267240611484004272 T^{6} + \)\(19\!\cdots\!84\)\( T^{7} + \)\(19\!\cdots\!07\)\( T^{8} + \)\(19\!\cdots\!84\)\( p^{5} T^{9} - 14267240611484004272 p^{10} T^{10} - 2604033925498692 p^{15} T^{11} + 9089867656090 p^{20} T^{12} + 565843800 p^{25} T^{13} - 4008500 p^{30} T^{14} + 276 p^{35} T^{15} + p^{40} T^{16} \)
19 \( 1 + 510 T - 3747425 T^{2} - 3937377630 T^{3} - 440016487199 T^{4} + 4115411338726080 T^{5} - 5704973911539330050 T^{6} + \)\(57\!\cdots\!00\)\( T^{7} + \)\(73\!\cdots\!50\)\( T^{8} + \)\(57\!\cdots\!00\)\( p^{5} T^{9} - 5704973911539330050 p^{10} T^{10} + 4115411338726080 p^{15} T^{11} - 440016487199 p^{20} T^{12} - 3937377630 p^{25} T^{13} - 3747425 p^{30} T^{14} + 510 p^{35} T^{15} + p^{40} T^{16} \)
23 \( 1 + 300 p T + 14742772 T^{2} - 5434854360 T^{3} - 64144356593702 T^{4} - 173260413144408420 T^{5} - \)\(23\!\cdots\!64\)\( T^{6} + \)\(19\!\cdots\!40\)\( T^{7} + \)\(99\!\cdots\!79\)\( T^{8} + \)\(19\!\cdots\!40\)\( p^{5} T^{9} - \)\(23\!\cdots\!64\)\( p^{10} T^{10} - 173260413144408420 p^{15} T^{11} - 64144356593702 p^{20} T^{12} - 5434854360 p^{25} T^{13} + 14742772 p^{30} T^{14} + 300 p^{36} T^{15} + p^{40} T^{16} \)
29 \( ( 1 - 540 T + 29394199 T^{2} + 29299685892 T^{3} + 772430149366772 T^{4} + 29299685892 p^{5} T^{5} + 29394199 p^{10} T^{6} - 540 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
31 \( 1 - 6410 T - 55714636 T^{2} + 363334551632 T^{3} + 2227342403022119 T^{4} - 10547841526395999412 T^{5} - \)\(75\!\cdots\!84\)\( T^{6} + \)\(10\!\cdots\!10\)\( T^{7} + \)\(25\!\cdots\!76\)\( T^{8} + \)\(10\!\cdots\!10\)\( p^{5} T^{9} - \)\(75\!\cdots\!84\)\( p^{10} T^{10} - 10547841526395999412 p^{15} T^{11} + 2227342403022119 p^{20} T^{12} + 363334551632 p^{25} T^{13} - 55714636 p^{30} T^{14} - 6410 p^{35} T^{15} + p^{40} T^{16} \)
37 \( 1 + 15250 T + 2042879 T^{2} + 164587155110 T^{3} + 277810073119073 p T^{4} - 25563809512056184540 T^{5} - \)\(73\!\cdots\!82\)\( T^{6} - \)\(25\!\cdots\!80\)\( T^{7} - \)\(18\!\cdots\!10\)\( T^{8} - \)\(25\!\cdots\!80\)\( p^{5} T^{9} - \)\(73\!\cdots\!82\)\( p^{10} T^{10} - 25563809512056184540 p^{15} T^{11} + 277810073119073 p^{21} T^{12} + 164587155110 p^{25} T^{13} + 2042879 p^{30} T^{14} + 15250 p^{35} T^{15} + p^{40} T^{16} \)
41 \( ( 1 - 4308 T + 270683560 T^{2} - 2598901038204 T^{3} + 34018560333944366 T^{4} - 2598901038204 p^{5} T^{5} + 270683560 p^{10} T^{6} - 4308 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
43 \( ( 1 - 29198 T + 787995265 T^{2} - 13076978932730 T^{3} + 187469286625894360 T^{4} - 13076978932730 p^{5} T^{5} + 787995265 p^{10} T^{6} - 29198 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
47 \( 1 - 15060 T - 741226172 T^{2} + 6055210491384 T^{3} + 443152946335783210 T^{4} - \)\(21\!\cdots\!28\)\( T^{5} - \)\(15\!\cdots\!68\)\( T^{6} + \)\(13\!\cdots\!48\)\( T^{7} + \)\(43\!\cdots\!71\)\( T^{8} + \)\(13\!\cdots\!48\)\( p^{5} T^{9} - \)\(15\!\cdots\!68\)\( p^{10} T^{10} - \)\(21\!\cdots\!28\)\( p^{15} T^{11} + 443152946335783210 p^{20} T^{12} + 6055210491384 p^{25} T^{13} - 741226172 p^{30} T^{14} - 15060 p^{35} T^{15} + p^{40} T^{16} \)
53 \( 1 + 13692 T - 987403679 T^{2} - 2191455111540 T^{3} + 630826063270172605 T^{4} - \)\(25\!\cdots\!24\)\( T^{5} - \)\(29\!\cdots\!74\)\( T^{6} + \)\(54\!\cdots\!28\)\( T^{7} + \)\(11\!\cdots\!06\)\( T^{8} + \)\(54\!\cdots\!28\)\( p^{5} T^{9} - \)\(29\!\cdots\!74\)\( p^{10} T^{10} - \)\(25\!\cdots\!24\)\( p^{15} T^{11} + 630826063270172605 p^{20} T^{12} - 2191455111540 p^{25} T^{13} - 987403679 p^{30} T^{14} + 13692 p^{35} T^{15} + p^{40} T^{16} \)
59 \( 1 + 34830 T - 1834824737 T^{2} - 42321414019590 T^{3} + 3370913750880697081 T^{4} + \)\(47\!\cdots\!20\)\( T^{5} - \)\(33\!\cdots\!82\)\( T^{6} - \)\(90\!\cdots\!20\)\( T^{7} + \)\(30\!\cdots\!74\)\( T^{8} - \)\(90\!\cdots\!20\)\( p^{5} T^{9} - \)\(33\!\cdots\!82\)\( p^{10} T^{10} + \)\(47\!\cdots\!20\)\( p^{15} T^{11} + 3370913750880697081 p^{20} T^{12} - 42321414019590 p^{25} T^{13} - 1834824737 p^{30} T^{14} + 34830 p^{35} T^{15} + p^{40} T^{16} \)
61 \( 1 - 5364 T - 2817016532 T^{2} + 5158230942312 T^{3} + 4645398449535186778 T^{4} - \)\(12\!\cdots\!24\)\( T^{5} - \)\(55\!\cdots\!52\)\( T^{6} + \)\(26\!\cdots\!00\)\( T^{7} + \)\(52\!\cdots\!87\)\( T^{8} + \)\(26\!\cdots\!00\)\( p^{5} T^{9} - \)\(55\!\cdots\!52\)\( p^{10} T^{10} - \)\(12\!\cdots\!24\)\( p^{15} T^{11} + 4645398449535186778 p^{20} T^{12} + 5158230942312 p^{25} T^{13} - 2817016532 p^{30} T^{14} - 5364 p^{35} T^{15} + p^{40} T^{16} \)
67 \( 1 - 5994 T - 2518577501 T^{2} + 43127704637370 T^{3} + 2548139821971114109 T^{4} - \)\(68\!\cdots\!04\)\( T^{5} + \)\(21\!\cdots\!62\)\( T^{6} + \)\(47\!\cdots\!72\)\( T^{7} - \)\(17\!\cdots\!02\)\( T^{8} + \)\(47\!\cdots\!72\)\( p^{5} T^{9} + \)\(21\!\cdots\!62\)\( p^{10} T^{10} - \)\(68\!\cdots\!04\)\( p^{15} T^{11} + 2548139821971114109 p^{20} T^{12} + 43127704637370 p^{25} T^{13} - 2518577501 p^{30} T^{14} - 5994 p^{35} T^{15} + p^{40} T^{16} \)
71 \( ( 1 - 89268 T + 8738662172 T^{2} - 466802240277492 T^{3} + 25044546792022133910 T^{4} - 466802240277492 p^{5} T^{5} + 8738662172 p^{10} T^{6} - 89268 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
73 \( 1 + 59638 T - 5098535509 T^{2} - 184967914738150 T^{3} + 26836978407770089433 T^{4} + \)\(52\!\cdots\!12\)\( T^{5} - \)\(81\!\cdots\!50\)\( T^{6} - \)\(30\!\cdots\!84\)\( T^{7} + \)\(20\!\cdots\!42\)\( T^{8} - \)\(30\!\cdots\!84\)\( p^{5} T^{9} - \)\(81\!\cdots\!50\)\( p^{10} T^{10} + \)\(52\!\cdots\!12\)\( p^{15} T^{11} + 26836978407770089433 p^{20} T^{12} - 184967914738150 p^{25} T^{13} - 5098535509 p^{30} T^{14} + 59638 p^{35} T^{15} + p^{40} T^{16} \)
79 \( 1 - 44062 T - 6584960844 T^{2} + 467510535225040 T^{3} + 20422039882069416887 T^{4} - \)\(19\!\cdots\!32\)\( T^{5} - \)\(53\!\cdots\!72\)\( T^{6} + \)\(30\!\cdots\!70\)\( T^{7} - \)\(76\!\cdots\!68\)\( T^{8} + \)\(30\!\cdots\!70\)\( p^{5} T^{9} - \)\(53\!\cdots\!72\)\( p^{10} T^{10} - \)\(19\!\cdots\!32\)\( p^{15} T^{11} + 20422039882069416887 p^{20} T^{12} + 467510535225040 p^{25} T^{13} - 6584960844 p^{30} T^{14} - 44062 p^{35} T^{15} + p^{40} T^{16} \)
83 \( ( 1 + 208446 T + 23363412401 T^{2} + 1724627286611514 T^{3} + \)\(11\!\cdots\!56\)\( T^{4} + 1724627286611514 p^{5} T^{5} + 23363412401 p^{10} T^{6} + 208446 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
89 \( 1 - 77520 T + 479326112 T^{2} - 1563841943328288 T^{3} + \)\(12\!\cdots\!70\)\( T^{4} - \)\(11\!\cdots\!72\)\( T^{5} + \)\(11\!\cdots\!40\)\( T^{6} - \)\(89\!\cdots\!20\)\( T^{7} + \)\(66\!\cdots\!67\)\( T^{8} - \)\(89\!\cdots\!20\)\( p^{5} T^{9} + \)\(11\!\cdots\!40\)\( p^{10} T^{10} - \)\(11\!\cdots\!72\)\( p^{15} T^{11} + \)\(12\!\cdots\!70\)\( p^{20} T^{12} - 1563841943328288 p^{25} T^{13} + 479326112 p^{30} T^{14} - 77520 p^{35} T^{15} + p^{40} T^{16} \)
97 \( ( 1 + 188630 T + 43620869129 T^{2} + 4760435860011510 T^{3} + \)\(59\!\cdots\!64\)\( T^{4} + 4760435860011510 p^{5} T^{5} + 43620869129 p^{10} T^{6} + 188630 p^{15} T^{7} + p^{20} T^{8} )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.113125646187576267229409539109, −7.76346036288667338088240328213, −7.38864140114291561487345493920, −7.33858803218183208350367612115, −7.11009184049610254783570598158, −6.74610556892610222850328816041, −6.58339595587481121657957889400, −6.17114696351992462762313902527, −6.16897695149997822842831244767, −5.73984646498827808965324466086, −5.68612600372883597512352142615, −5.50981562690680467665656021301, −5.35191017856911452163792293594, −4.67259239994761342123715478166, −4.60107374630584079775977856462, −4.40540228070355111938452123360, −4.14103377745463855692439433648, −3.68650261413003741380346724003, −3.00472761016487634143420299192, −2.59835994850349768932952320178, −2.36299147365530659116393497233, −1.55016034279135827785323409561, −1.50596623695673060604327933545, −0.863360223626273806926024241837, −0.28155428817559639907044665997, 0.28155428817559639907044665997, 0.863360223626273806926024241837, 1.50596623695673060604327933545, 1.55016034279135827785323409561, 2.36299147365530659116393497233, 2.59835994850349768932952320178, 3.00472761016487634143420299192, 3.68650261413003741380346724003, 4.14103377745463855692439433648, 4.40540228070355111938452123360, 4.60107374630584079775977856462, 4.67259239994761342123715478166, 5.35191017856911452163792293594, 5.50981562690680467665656021301, 5.68612600372883597512352142615, 5.73984646498827808965324466086, 6.16897695149997822842831244767, 6.17114696351992462762313902527, 6.58339595587481121657957889400, 6.74610556892610222850328816041, 7.11009184049610254783570598158, 7.33858803218183208350367612115, 7.38864140114291561487345493920, 7.76346036288667338088240328213, 8.113125646187576267229409539109

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

Plot not available for L-functions of degree greater than 10.