Dirichlet series
| L(s) = 1 | − 84·3-s + 3.75e3·5-s − 1.44e4·7-s − 4.11e4·9-s − 7.23e3·11-s − 1.28e5·13-s − 3.15e5·15-s + 7.82e3·17-s + 1.08e6·19-s + 1.21e6·21-s − 4.33e5·23-s + 8.20e6·25-s + 2.34e6·27-s − 5.03e6·29-s + 8.00e6·31-s + 6.07e5·33-s − 5.40e7·35-s − 3.17e6·37-s + 1.07e7·39-s − 4.31e7·41-s + 1.71e7·43-s − 1.54e8·45-s + 1.59e7·47-s + 1.21e8·49-s − 6.57e5·51-s − 1.99e8·53-s − 2.71e7·55-s + ⋯ |
| L(s) = 1 | − 0.598·3-s + 2.68·5-s − 2.26·7-s − 2.09·9-s − 0.149·11-s − 1.24·13-s − 1.60·15-s + 0.0227·17-s + 1.91·19-s + 1.35·21-s − 0.323·23-s + 21/5·25-s + 0.849·27-s − 1.32·29-s + 1.55·31-s + 0.0892·33-s − 6.08·35-s − 0.278·37-s + 0.746·39-s − 2.38·41-s + 0.763·43-s − 5.61·45-s + 0.476·47-s + 3·49-s − 0.0136·51-s − 3.46·53-s − 0.399·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{18} \cdot 5^{6} \cdot 7^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.99441\times 10^{12}\) |
| Root analytic conductor: | \(12.0087\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 2^{18} \cdot 5^{6} \cdot 7^{6} ,\ ( \ : [9/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - p^{4} T )^{6} \) | |
| 7 | \( ( 1 + p^{4} T )^{6} \) | |
| good | 3 | \( 1 + 28 p T + 48224 T^{2} + 573640 p^{2} T^{3} + 143550872 p^{2} T^{4} + 2369992028 p^{4} T^{5} + 361804250870 p^{4} T^{6} + 2369992028 p^{13} T^{7} + 143550872 p^{20} T^{8} + 573640 p^{29} T^{9} + 48224 p^{36} T^{10} + 28 p^{46} T^{11} + p^{54} T^{12} \) |
| 11 | \( 1 + 7236 T + 7695504440 T^{2} + 8619106210424 p T^{3} + 33062670125192984416 T^{4} + \)\(36\!\cdots\!52\)\( T^{5} + \)\(96\!\cdots\!74\)\( T^{6} + \)\(36\!\cdots\!52\)\( p^{9} T^{7} + 33062670125192984416 p^{18} T^{8} + 8619106210424 p^{28} T^{9} + 7695504440 p^{36} T^{10} + 7236 p^{45} T^{11} + p^{54} T^{12} \) | |
| 13 | \( 1 + 128320 T + 43437799360 T^{2} + 2858376523705108 T^{3} + 49830988794716511168 p T^{4} + \)\(17\!\cdots\!72\)\( T^{5} + \)\(63\!\cdots\!18\)\( T^{6} + \)\(17\!\cdots\!72\)\( p^{9} T^{7} + 49830988794716511168 p^{19} T^{8} + 2858376523705108 p^{27} T^{9} + 43437799360 p^{36} T^{10} + 128320 p^{45} T^{11} + p^{54} T^{12} \) | |
| 17 | \( 1 - 7824 T + 234437453832 T^{2} - 3440167854936020 T^{3} + \)\(30\!\cdots\!20\)\( T^{4} + \)\(28\!\cdots\!24\)\( p T^{5} + \)\(40\!\cdots\!58\)\( T^{6} + \)\(28\!\cdots\!24\)\( p^{10} T^{7} + \)\(30\!\cdots\!20\)\( p^{18} T^{8} - 3440167854936020 p^{27} T^{9} + 234437453832 p^{36} T^{10} - 7824 p^{45} T^{11} + p^{54} T^{12} \) | |
| 19 | \( 1 - 1086424 T + 1532568526778 T^{2} - 830482693146918568 T^{3} + \)\(62\!\cdots\!95\)\( T^{4} - \)\(16\!\cdots\!36\)\( T^{5} + \)\(14\!\cdots\!48\)\( T^{6} - \)\(16\!\cdots\!36\)\( p^{9} T^{7} + \)\(62\!\cdots\!95\)\( p^{18} T^{8} - 830482693146918568 p^{27} T^{9} + 1532568526778 p^{36} T^{10} - 1086424 p^{45} T^{11} + p^{54} T^{12} \) | |
| 23 | \( 1 + 433840 T + 8144727354978 T^{2} + 3433924433131033360 T^{3} + \)\(30\!\cdots\!35\)\( T^{4} + \)\(11\!\cdots\!28\)\( T^{5} + \)\(70\!\cdots\!40\)\( T^{6} + \)\(11\!\cdots\!28\)\( p^{9} T^{7} + \)\(30\!\cdots\!35\)\( p^{18} T^{8} + 3433924433131033360 p^{27} T^{9} + 8144727354978 p^{36} T^{10} + 433840 p^{45} T^{11} + p^{54} T^{12} \) | |
| 29 | \( 1 + 5035320 T + 37853342404664 T^{2} + 62429128817164606644 T^{3} + \)\(64\!\cdots\!36\)\( T^{4} + \)\(17\!\cdots\!64\)\( T^{5} + \)\(15\!\cdots\!26\)\( T^{6} + \)\(17\!\cdots\!64\)\( p^{9} T^{7} + \)\(64\!\cdots\!36\)\( p^{18} T^{8} + 62429128817164606644 p^{27} T^{9} + 37853342404664 p^{36} T^{10} + 5035320 p^{45} T^{11} + p^{54} T^{12} \) | |
| 31 | \( 1 - 8009528 T + 119205133544106 T^{2} - \)\(62\!\cdots\!20\)\( T^{3} + \)\(54\!\cdots\!95\)\( T^{4} - \)\(21\!\cdots\!44\)\( T^{5} + \)\(15\!\cdots\!72\)\( T^{6} - \)\(21\!\cdots\!44\)\( p^{9} T^{7} + \)\(54\!\cdots\!95\)\( p^{18} T^{8} - \)\(62\!\cdots\!20\)\( p^{27} T^{9} + 119205133544106 p^{36} T^{10} - 8009528 p^{45} T^{11} + p^{54} T^{12} \) | |
| 37 | \( 1 + 3174020 T + 670943833183482 T^{2} + \)\(20\!\cdots\!48\)\( T^{3} + \)\(19\!\cdots\!31\)\( T^{4} + \)\(53\!\cdots\!36\)\( T^{5} + \)\(33\!\cdots\!36\)\( T^{6} + \)\(53\!\cdots\!36\)\( p^{9} T^{7} + \)\(19\!\cdots\!31\)\( p^{18} T^{8} + \)\(20\!\cdots\!48\)\( p^{27} T^{9} + 670943833183482 p^{36} T^{10} + 3174020 p^{45} T^{11} + p^{54} T^{12} \) | |
| 41 | \( 1 + 43110164 T + 1804437888553066 T^{2} + \)\(48\!\cdots\!44\)\( T^{3} + \)\(13\!\cdots\!59\)\( T^{4} + \)\(27\!\cdots\!32\)\( T^{5} + \)\(55\!\cdots\!88\)\( T^{6} + \)\(27\!\cdots\!32\)\( p^{9} T^{7} + \)\(13\!\cdots\!59\)\( p^{18} T^{8} + \)\(48\!\cdots\!44\)\( p^{27} T^{9} + 1804437888553066 p^{36} T^{10} + 43110164 p^{45} T^{11} + p^{54} T^{12} \) | |
| 43 | \( 1 - 17127272 T + 2044129123278986 T^{2} - \)\(28\!\cdots\!80\)\( T^{3} + \)\(20\!\cdots\!59\)\( T^{4} - \)\(24\!\cdots\!00\)\( T^{5} + \)\(13\!\cdots\!60\)\( T^{6} - \)\(24\!\cdots\!00\)\( p^{9} T^{7} + \)\(20\!\cdots\!59\)\( p^{18} T^{8} - \)\(28\!\cdots\!80\)\( p^{27} T^{9} + 2044129123278986 p^{36} T^{10} - 17127272 p^{45} T^{11} + p^{54} T^{12} \) | |
| 47 | \( 1 - 15939092 T + 4455698275804552 T^{2} - \)\(67\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!60\)\( T^{4} - \)\(13\!\cdots\!04\)\( T^{5} + \)\(14\!\cdots\!18\)\( T^{6} - \)\(13\!\cdots\!04\)\( p^{9} T^{7} + \)\(10\!\cdots\!60\)\( p^{18} T^{8} - \)\(67\!\cdots\!00\)\( p^{27} T^{9} + 4455698275804552 p^{36} T^{10} - 15939092 p^{45} T^{11} + p^{54} T^{12} \) | |
| 53 | \( 1 + 199005812 T + 28309855541510626 T^{2} + \)\(26\!\cdots\!24\)\( T^{3} + \)\(21\!\cdots\!31\)\( T^{4} + \)\(13\!\cdots\!76\)\( T^{5} + \)\(83\!\cdots\!92\)\( T^{6} + \)\(13\!\cdots\!76\)\( p^{9} T^{7} + \)\(21\!\cdots\!31\)\( p^{18} T^{8} + \)\(26\!\cdots\!24\)\( p^{27} T^{9} + 28309855541510626 p^{36} T^{10} + 199005812 p^{45} T^{11} + p^{54} T^{12} \) | |
| 59 | \( 1 + 17573560 T + 36786451275451794 T^{2} + \)\(95\!\cdots\!00\)\( T^{3} + \)\(62\!\cdots\!75\)\( T^{4} + \)\(18\!\cdots\!04\)\( T^{5} + \)\(66\!\cdots\!40\)\( T^{6} + \)\(18\!\cdots\!04\)\( p^{9} T^{7} + \)\(62\!\cdots\!75\)\( p^{18} T^{8} + \)\(95\!\cdots\!00\)\( p^{27} T^{9} + 36786451275451794 p^{36} T^{10} + 17573560 p^{45} T^{11} + p^{54} T^{12} \) | |
| 61 | \( 1 + 45679132 T + 41984396849942850 T^{2} + \)\(32\!\cdots\!96\)\( T^{3} + \)\(89\!\cdots\!35\)\( T^{4} + \)\(75\!\cdots\!80\)\( T^{5} + \)\(12\!\cdots\!68\)\( T^{6} + \)\(75\!\cdots\!80\)\( p^{9} T^{7} + \)\(89\!\cdots\!35\)\( p^{18} T^{8} + \)\(32\!\cdots\!96\)\( p^{27} T^{9} + 41984396849942850 p^{36} T^{10} + 45679132 p^{45} T^{11} + p^{54} T^{12} \) | |
| 67 | \( 1 - 368557032 T + 123735080493545570 T^{2} - \)\(30\!\cdots\!20\)\( T^{3} + \)\(68\!\cdots\!47\)\( T^{4} - \)\(12\!\cdots\!40\)\( T^{5} + \)\(23\!\cdots\!04\)\( T^{6} - \)\(12\!\cdots\!40\)\( p^{9} T^{7} + \)\(68\!\cdots\!47\)\( p^{18} T^{8} - \)\(30\!\cdots\!20\)\( p^{27} T^{9} + 123735080493545570 p^{36} T^{10} - 368557032 p^{45} T^{11} + p^{54} T^{12} \) | |
| 71 | \( 1 + 117886480 T + 198795796451173098 T^{2} + \)\(33\!\cdots\!68\)\( T^{3} + \)\(17\!\cdots\!63\)\( T^{4} + \)\(32\!\cdots\!24\)\( T^{5} + \)\(97\!\cdots\!32\)\( T^{6} + \)\(32\!\cdots\!24\)\( p^{9} T^{7} + \)\(17\!\cdots\!63\)\( p^{18} T^{8} + \)\(33\!\cdots\!68\)\( p^{27} T^{9} + 198795796451173098 p^{36} T^{10} + 117886480 p^{45} T^{11} + p^{54} T^{12} \) | |
| 73 | \( 1 + 497207908 T + 337179448231469746 T^{2} + \)\(11\!\cdots\!96\)\( T^{3} + \)\(48\!\cdots\!11\)\( T^{4} + \)\(12\!\cdots\!84\)\( T^{5} + \)\(37\!\cdots\!32\)\( T^{6} + \)\(12\!\cdots\!84\)\( p^{9} T^{7} + \)\(48\!\cdots\!11\)\( p^{18} T^{8} + \)\(11\!\cdots\!96\)\( p^{27} T^{9} + 337179448231469746 p^{36} T^{10} + 497207908 p^{45} T^{11} + p^{54} T^{12} \) | |
| 79 | \( 1 - 366960940 T + 93272210216677424 T^{2} - \)\(63\!\cdots\!96\)\( T^{3} + \)\(37\!\cdots\!56\)\( T^{4} - \)\(10\!\cdots\!96\)\( T^{5} + \)\(35\!\cdots\!66\)\( T^{6} - \)\(10\!\cdots\!96\)\( p^{9} T^{7} + \)\(37\!\cdots\!56\)\( p^{18} T^{8} - \)\(63\!\cdots\!96\)\( p^{27} T^{9} + 93272210216677424 p^{36} T^{10} - 366960940 p^{45} T^{11} + p^{54} T^{12} \) | |
| 83 | \( 1 + 114133272 T + 451072395338516258 T^{2} + \)\(26\!\cdots\!00\)\( T^{3} + \)\(10\!\cdots\!15\)\( T^{4} - \)\(24\!\cdots\!44\)\( T^{5} + \)\(20\!\cdots\!72\)\( T^{6} - \)\(24\!\cdots\!44\)\( p^{9} T^{7} + \)\(10\!\cdots\!15\)\( p^{18} T^{8} + \)\(26\!\cdots\!00\)\( p^{27} T^{9} + 451072395338516258 p^{36} T^{10} + 114133272 p^{45} T^{11} + p^{54} T^{12} \) | |
| 89 | \( 1 + 522324452 T + 458675478332366474 T^{2} + \)\(11\!\cdots\!24\)\( T^{3} + \)\(75\!\cdots\!35\)\( T^{4} + \)\(16\!\cdots\!28\)\( T^{5} + \)\(79\!\cdots\!00\)\( T^{6} + \)\(16\!\cdots\!28\)\( p^{9} T^{7} + \)\(75\!\cdots\!35\)\( p^{18} T^{8} + \)\(11\!\cdots\!24\)\( p^{27} T^{9} + 458675478332366474 p^{36} T^{10} + 522324452 p^{45} T^{11} + p^{54} T^{12} \) | |
| 97 | \( 1 + 1013121248 T + 2256014264527231544 T^{2} + \)\(21\!\cdots\!72\)\( T^{3} + \)\(33\!\cdots\!16\)\( T^{4} + \)\(25\!\cdots\!96\)\( p T^{5} + \)\(30\!\cdots\!30\)\( T^{6} + \)\(25\!\cdots\!96\)\( p^{10} T^{7} + \)\(33\!\cdots\!16\)\( p^{18} T^{8} + \)\(21\!\cdots\!72\)\( p^{27} T^{9} + 2256014264527231544 p^{36} T^{10} + 1013121248 p^{45} T^{11} + p^{54} T^{12} \) | |
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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
−5.58404919267583424556661181665, −5.24648313889159800587468241216, −5.04366871038567536730401448208, −5.01352005805877311512289208637, −4.81996546672436511131846418535, −4.80943085324243348805985876880, −4.47073982667698129372810877532, −4.03564616923602936729646306269, −3.64050384869354682021175304761, −3.59555716583351879047245507115, −3.54416695133348696496155915595, −3.45565040071558817113394153346, −3.10054984921253354265381414748, −2.84012935394196805137593095977, −2.60643718463675674893977347443, −2.49925049591699966590450486175, −2.46320820022389964091809327062, −2.44117157931671032857119602659, −2.21443835073361552613859301927, −1.66279226203630807321674696250, −1.51204335778588098341701476596, −1.24330916120365187350911337809, −1.17121162674775003810391098430, −1.05585145931338904057702389230, −0.941833732899086560487953526454, 0, 0, 0, 0, 0, 0, 0.941833732899086560487953526454, 1.05585145931338904057702389230, 1.17121162674775003810391098430, 1.24330916120365187350911337809, 1.51204335778588098341701476596, 1.66279226203630807321674696250, 2.21443835073361552613859301927, 2.44117157931671032857119602659, 2.46320820022389964091809327062, 2.49925049591699966590450486175, 2.60643718463675674893977347443, 2.84012935394196805137593095977, 3.10054984921253354265381414748, 3.45565040071558817113394153346, 3.54416695133348696496155915595, 3.59555716583351879047245507115, 3.64050384869354682021175304761, 4.03564616923602936729646306269, 4.47073982667698129372810877532, 4.80943085324243348805985876880, 4.81996546672436511131846418535, 5.01352005805877311512289208637, 5.04366871038567536730401448208, 5.24648313889159800587468241216, 5.58404919267583424556661181665
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.