Dirichlet series
| L(s) = 1 | + 48·2-s + 768·4-s + 733·5-s + 5.01e3·7-s − 8.19e3·8-s + 3.51e4·10-s − 7.33e3·11-s + 1.97e5·13-s + 2.40e5·14-s − 5.89e5·16-s + 3.06e5·17-s − 3.77e5·19-s + 5.62e5·20-s − 3.52e5·22-s + 2.26e6·23-s + 3.12e6·25-s + 9.45e6·26-s + 3.84e6·28-s + 1.30e7·29-s − 6.65e6·31-s − 9.43e6·32-s + 1.47e7·34-s + 3.67e6·35-s − 2.22e7·37-s − 1.81e7·38-s − 6.00e6·40-s − 6.80e7·41-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 3/2·4-s + 0.524·5-s + 0.788·7-s − 0.707·8-s + 1.11·10-s − 0.151·11-s + 1.91·13-s + 1.67·14-s − 9/4·16-s + 0.890·17-s − 0.665·19-s + 0.786·20-s − 0.320·22-s + 1.68·23-s + 1.60·25-s + 4.05·26-s + 1.18·28-s + 3.43·29-s − 1.29·31-s − 1.59·32-s + 1.88·34-s + 0.413·35-s − 1.95·37-s − 1.41·38-s − 0.370·40-s − 3.76·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{12} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 3^{12} \cdot 7^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.46875\times 10^{10}\) |
| Root analytic conductor: | \(8.05571\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{6} \cdot 3^{12} \cdot 7^{6} ,\ ( \ : [9/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(12.09313136\) |
| \(L(\frac12)\) | \(\approx\) | \(12.09313136\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{4} T + p^{8} T^{2} )^{3} \) |
| 3 | \( 1 \) | |
| 7 | \( 1 - 716 p T - 6763 p^{4} T^{2} + 132184 p^{7} T^{3} - 6763 p^{13} T^{4} - 716 p^{19} T^{5} + p^{27} T^{6} \) | |
| good | 5 | \( 1 - 733 T - 2589737 T^{2} - 1661777128 T^{3} + 3439801929581 T^{4} + 1034369617595257 p T^{5} - 245893183000998026 p^{2} T^{6} + 1034369617595257 p^{10} T^{7} + 3439801929581 p^{18} T^{8} - 1661777128 p^{27} T^{9} - 2589737 p^{36} T^{10} - 733 p^{45} T^{11} + p^{54} T^{12} \) |
| 11 | \( 1 + 7339 T - 1638063011 T^{2} + 59929131803902 T^{3} - 86592352599819335 p T^{4} - \)\(57\!\cdots\!41\)\( T^{5} + \)\(19\!\cdots\!86\)\( T^{6} - \)\(57\!\cdots\!41\)\( p^{9} T^{7} - 86592352599819335 p^{19} T^{8} + 59929131803902 p^{27} T^{9} - 1638063011 p^{36} T^{10} + 7339 p^{45} T^{11} + p^{54} T^{12} \) | |
| 13 | \( ( 1 - 98518 T + 23128160899 T^{2} - 2151914079092708 T^{3} + 23128160899 p^{9} T^{4} - 98518 p^{18} T^{5} + p^{27} T^{6} )^{2} \) | |
| 17 | \( 1 - 306665 T - 118249479533 T^{2} + 40915218770279116 T^{3} + \)\(36\!\cdots\!93\)\( T^{4} - \)\(12\!\cdots\!99\)\( T^{5} - \)\(52\!\cdots\!38\)\( T^{6} - \)\(12\!\cdots\!99\)\( p^{9} T^{7} + \)\(36\!\cdots\!93\)\( p^{18} T^{8} + 40915218770279116 p^{27} T^{9} - 118249479533 p^{36} T^{10} - 306665 p^{45} T^{11} + p^{54} T^{12} \) | |
| 19 | \( 1 + 377991 T - 527326979931 T^{2} - 307112754046828810 T^{3} + \)\(12\!\cdots\!55\)\( T^{4} + \)\(63\!\cdots\!07\)\( T^{5} - \)\(14\!\cdots\!18\)\( T^{6} + \)\(63\!\cdots\!07\)\( p^{9} T^{7} + \)\(12\!\cdots\!55\)\( p^{18} T^{8} - 307112754046828810 p^{27} T^{9} - 527326979931 p^{36} T^{10} + 377991 p^{45} T^{11} + p^{54} T^{12} \) | |
| 23 | \( 1 - 2267255 T - 572332935647 T^{2} + 3259880926523493262 T^{3} + \)\(39\!\cdots\!43\)\( T^{4} - \)\(70\!\cdots\!27\)\( T^{5} + \)\(19\!\cdots\!98\)\( T^{6} - \)\(70\!\cdots\!27\)\( p^{9} T^{7} + \)\(39\!\cdots\!43\)\( p^{18} T^{8} + 3259880926523493262 p^{27} T^{9} - 572332935647 p^{36} T^{10} - 2267255 p^{45} T^{11} + p^{54} T^{12} \) | |
| 29 | \( ( 1 - 6542978 T + 27077371504275 T^{2} - 2337075386063582620 p T^{3} + 27077371504275 p^{9} T^{4} - 6542978 p^{18} T^{5} + p^{27} T^{6} )^{2} \) | |
| 31 | \( 1 + 6654517 T - 35266205130495 T^{2} - 3416211213892327710 p T^{3} + \)\(21\!\cdots\!19\)\( T^{4} + \)\(19\!\cdots\!17\)\( T^{5} - \)\(60\!\cdots\!42\)\( T^{6} + \)\(19\!\cdots\!17\)\( p^{9} T^{7} + \)\(21\!\cdots\!19\)\( p^{18} T^{8} - 3416211213892327710 p^{28} T^{9} - 35266205130495 p^{36} T^{10} + 6654517 p^{45} T^{11} + p^{54} T^{12} \) | |
| 37 | \( 1 + 22287969 T - 21443869395225 T^{2} - \)\(46\!\cdots\!72\)\( T^{3} + \)\(67\!\cdots\!17\)\( T^{4} + \)\(48\!\cdots\!23\)\( T^{5} - \)\(28\!\cdots\!86\)\( T^{6} + \)\(48\!\cdots\!23\)\( p^{9} T^{7} + \)\(67\!\cdots\!17\)\( p^{18} T^{8} - \)\(46\!\cdots\!72\)\( p^{27} T^{9} - 21443869395225 p^{36} T^{10} + 22287969 p^{45} T^{11} + p^{54} T^{12} \) | |
| 41 | \( ( 1 + 34048098 T + 1300690668411159 T^{2} + \)\(23\!\cdots\!64\)\( T^{3} + 1300690668411159 p^{9} T^{4} + 34048098 p^{18} T^{5} + p^{27} T^{6} )^{2} \) | |
| 43 | \( ( 1 + 62824140 T + 2320736417731089 T^{2} + \)\(57\!\cdots\!40\)\( T^{3} + 2320736417731089 p^{9} T^{4} + 62824140 p^{18} T^{5} + p^{27} T^{6} )^{2} \) | |
| 47 | \( 1 - 52703019 T - 1338372320657487 T^{2} + \)\(23\!\cdots\!74\)\( T^{3} + \)\(59\!\cdots\!71\)\( T^{4} - \)\(75\!\cdots\!47\)\( T^{5} - \)\(51\!\cdots\!22\)\( T^{6} - \)\(75\!\cdots\!47\)\( p^{9} T^{7} + \)\(59\!\cdots\!71\)\( p^{18} T^{8} + \)\(23\!\cdots\!74\)\( p^{27} T^{9} - 1338372320657487 p^{36} T^{10} - 52703019 p^{45} T^{11} + p^{54} T^{12} \) | |
| 53 | \( 1 - 12091125 T - 3093729837293337 T^{2} - \)\(34\!\cdots\!48\)\( T^{3} + \)\(16\!\cdots\!73\)\( T^{4} + \)\(62\!\cdots\!13\)\( T^{5} + \)\(59\!\cdots\!98\)\( T^{6} + \)\(62\!\cdots\!13\)\( p^{9} T^{7} + \)\(16\!\cdots\!73\)\( p^{18} T^{8} - \)\(34\!\cdots\!48\)\( p^{27} T^{9} - 3093729837293337 p^{36} T^{10} - 12091125 p^{45} T^{11} + p^{54} T^{12} \) | |
| 59 | \( 1 - 12949897 T - 1594882595738603 T^{2} + \)\(30\!\cdots\!78\)\( T^{3} - \)\(32\!\cdots\!61\)\( T^{4} - \)\(32\!\cdots\!97\)\( T^{5} + \)\(36\!\cdots\!74\)\( T^{6} - \)\(32\!\cdots\!97\)\( p^{9} T^{7} - \)\(32\!\cdots\!61\)\( p^{18} T^{8} + \)\(30\!\cdots\!78\)\( p^{27} T^{9} - 1594882595738603 p^{36} T^{10} - 12949897 p^{45} T^{11} + p^{54} T^{12} \) | |
| 61 | \( 1 + 160252153 T - 14234381709867777 T^{2} - \)\(10\!\cdots\!56\)\( T^{3} + \)\(53\!\cdots\!49\)\( T^{4} + \)\(32\!\cdots\!99\)\( p T^{5} - \)\(14\!\cdots\!94\)\( p^{2} T^{6} + \)\(32\!\cdots\!99\)\( p^{10} T^{7} + \)\(53\!\cdots\!49\)\( p^{18} T^{8} - \)\(10\!\cdots\!56\)\( p^{27} T^{9} - 14234381709867777 p^{36} T^{10} + 160252153 p^{45} T^{11} + p^{54} T^{12} \) | |
| 67 | \( 1 + 480890225 T + 74976954078885197 T^{2} + \)\(15\!\cdots\!34\)\( T^{3} + \)\(58\!\cdots\!43\)\( T^{4} + \)\(13\!\cdots\!47\)\( p T^{5} + \)\(96\!\cdots\!62\)\( T^{6} + \)\(13\!\cdots\!47\)\( p^{10} T^{7} + \)\(58\!\cdots\!43\)\( p^{18} T^{8} + \)\(15\!\cdots\!34\)\( p^{27} T^{9} + 74976954078885197 p^{36} T^{10} + 480890225 p^{45} T^{11} + p^{54} T^{12} \) | |
| 71 | \( ( 1 - 37210720 T + 117772904777029461 T^{2} - \)\(37\!\cdots\!56\)\( T^{3} + 117772904777029461 p^{9} T^{4} - 37210720 p^{18} T^{5} + p^{27} T^{6} )^{2} \) | |
| 73 | \( 1 - 251382283 T - 16115949267395701 T^{2} + \)\(40\!\cdots\!76\)\( T^{3} - \)\(60\!\cdots\!79\)\( T^{4} - \)\(93\!\cdots\!77\)\( T^{5} + \)\(74\!\cdots\!82\)\( T^{6} - \)\(93\!\cdots\!77\)\( p^{9} T^{7} - \)\(60\!\cdots\!79\)\( p^{18} T^{8} + \)\(40\!\cdots\!76\)\( p^{27} T^{9} - 16115949267395701 p^{36} T^{10} - 251382283 p^{45} T^{11} + p^{54} T^{12} \) | |
| 79 | \( 1 - 286494785 T - 228249178975019367 T^{2} + \)\(51\!\cdots\!90\)\( T^{3} + \)\(39\!\cdots\!91\)\( T^{4} - \)\(45\!\cdots\!85\)\( T^{5} - \)\(43\!\cdots\!98\)\( T^{6} - \)\(45\!\cdots\!85\)\( p^{9} T^{7} + \)\(39\!\cdots\!91\)\( p^{18} T^{8} + \)\(51\!\cdots\!90\)\( p^{27} T^{9} - 228249178975019367 p^{36} T^{10} - 286494785 p^{45} T^{11} + p^{54} T^{12} \) | |
| 83 | \( ( 1 + 1147591172 T + 974458842151282569 T^{2} + \)\(47\!\cdots\!96\)\( T^{3} + 974458842151282569 p^{9} T^{4} + 1147591172 p^{18} T^{5} + p^{27} T^{6} )^{2} \) | |
| 89 | \( 1 - 901243845 T - 346930495326018789 T^{2} + \)\(12\!\cdots\!16\)\( T^{3} + \)\(41\!\cdots\!85\)\( T^{4} - \)\(56\!\cdots\!27\)\( T^{5} - \)\(14\!\cdots\!02\)\( T^{6} - \)\(56\!\cdots\!27\)\( p^{9} T^{7} + \)\(41\!\cdots\!85\)\( p^{18} T^{8} + \)\(12\!\cdots\!16\)\( p^{27} T^{9} - 346930495326018789 p^{36} T^{10} - 901243845 p^{45} T^{11} + p^{54} T^{12} \) | |
| 97 | \( ( 1 - 314853938 T + 758253851848651967 T^{2} - \)\(23\!\cdots\!12\)\( T^{3} + 758253851848651967 p^{9} T^{4} - 314853938 p^{18} T^{5} + p^{27} T^{6} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.48094298466445790298803911622, −5.41849525499665915032438450186, −5.11947901333839728213831771820, −5.08558199787236175215388023820, −4.70725707374689554387308220260, −4.68065868834119980405918693038, −4.55542177959438284906499480086, −4.36063241004809830136272328904, −3.73509588651654653651349841158, −3.65014567770396976771052107979, −3.55338660080365739346291250993, −3.49998054068044999891892531051, −3.04923697275597484965649042564, −2.91062084525451885836255971453, −2.87935625282039030448618078655, −2.58819031029701289823105736568, −2.16376223986556262778932584719, −1.63824786750457564845137870153, −1.55307564284552754229837915073, −1.52118165958034197256490919530, −1.41686236897119728163754614150, −1.04572018210166152356983187764, −0.58915054794039566337415106321, −0.46262671012939931783993335073, −0.15071191702529597369446032844, 0.15071191702529597369446032844, 0.46262671012939931783993335073, 0.58915054794039566337415106321, 1.04572018210166152356983187764, 1.41686236897119728163754614150, 1.52118165958034197256490919530, 1.55307564284552754229837915073, 1.63824786750457564845137870153, 2.16376223986556262778932584719, 2.58819031029701289823105736568, 2.87935625282039030448618078655, 2.91062084525451885836255971453, 3.04923697275597484965649042564, 3.49998054068044999891892531051, 3.55338660080365739346291250993, 3.65014567770396976771052107979, 3.73509588651654653651349841158, 4.36063241004809830136272328904, 4.55542177959438284906499480086, 4.68065868834119980405918693038, 4.70725707374689554387308220260, 5.08558199787236175215388023820, 5.11947901333839728213831771820, 5.41849525499665915032438450186, 5.48094298466445790298803911622
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.