Dirichlet series
| L(s) = 1 | − 2-s + 2·4-s + 30·5-s + 47·7-s − 16·8-s − 30·10-s − 81·11-s + 169·13-s − 47·14-s − 53·16-s + 17-s + 116·19-s + 60·20-s + 81·22-s + 52·23-s + 525·25-s − 169·26-s + 94·28-s − 174·29-s + 340·31-s + 21·32-s − 34-s + 1.41e3·35-s + 332·37-s − 116·38-s − 480·40-s + 616·41-s + ⋯ |
| L(s) = 1 | − 0.353·2-s + 1/4·4-s + 2.68·5-s + 2.53·7-s − 0.707·8-s − 0.948·10-s − 2.22·11-s + 3.60·13-s − 0.897·14-s − 0.828·16-s + 0.0142·17-s + 1.40·19-s + 0.670·20-s + 0.784·22-s + 0.471·23-s + 21/5·25-s − 1.27·26-s + 0.634·28-s − 1.11·29-s + 1.96·31-s + 0.116·32-s − 0.00504·34-s + 6.80·35-s + 1.47·37-s − 0.495·38-s − 1.89·40-s + 2.34·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{6} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(3^{12} \cdot 5^{6} \cdot 29^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.08381\times 10^{11}\) |
| Root analytic conductor: | \(8.77482\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 3^{12} \cdot 5^{6} \cdot 29^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(138.5304385\) |
| \(L(\frac12)\) | \(\approx\) | \(138.5304385\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 - p T )^{6} \) | |
| 29 | \( ( 1 + p T )^{6} \) | |
| good | 2 | \( 1 + T - T^{2} + 13 T^{3} + 21 p^{2} T^{4} + 37 p T^{5} - 23 p^{2} T^{6} + 37 p^{4} T^{7} + 21 p^{8} T^{8} + 13 p^{9} T^{9} - p^{12} T^{10} + p^{15} T^{11} + p^{18} T^{12} \) |
| 7 | \( 1 - 47 T + 242 p T^{2} - 48633 T^{3} + 1269399 T^{4} - 27084898 T^{5} + 536364308 T^{6} - 27084898 p^{3} T^{7} + 1269399 p^{6} T^{8} - 48633 p^{9} T^{9} + 242 p^{13} T^{10} - 47 p^{15} T^{11} + p^{18} T^{12} \) | |
| 11 | \( 1 + 81 T + 7285 T^{2} + 400470 T^{3} + 21225287 T^{4} + 884246505 T^{5} + 35457827646 T^{6} + 884246505 p^{3} T^{7} + 21225287 p^{6} T^{8} + 400470 p^{9} T^{9} + 7285 p^{12} T^{10} + 81 p^{15} T^{11} + p^{18} T^{12} \) | |
| 13 | \( 1 - p^{2} T + 22166 T^{2} - 2003717 T^{3} + 150322519 T^{4} - 9100077590 T^{5} + 467902766548 T^{6} - 9100077590 p^{3} T^{7} + 150322519 p^{6} T^{8} - 2003717 p^{9} T^{9} + 22166 p^{12} T^{10} - p^{17} T^{11} + p^{18} T^{12} \) | |
| 17 | \( 1 - T + 22636 T^{2} + 54303 T^{3} + 231271391 T^{4} + 914499890 T^{5} + 1416714172968 T^{6} + 914499890 p^{3} T^{7} + 231271391 p^{6} T^{8} + 54303 p^{9} T^{9} + 22636 p^{12} T^{10} - p^{15} T^{11} + p^{18} T^{12} \) | |
| 19 | \( 1 - 116 T + 31902 T^{2} - 2590388 T^{3} + 453795591 T^{4} - 30145107120 T^{5} + 209154567628 p T^{6} - 30145107120 p^{3} T^{7} + 453795591 p^{6} T^{8} - 2590388 p^{9} T^{9} + 31902 p^{12} T^{10} - 116 p^{15} T^{11} + p^{18} T^{12} \) | |
| 23 | \( 1 - 52 T + 24683 T^{2} - 1496510 T^{3} + 282119759 T^{4} - 27844771778 T^{5} + 2488866804234 T^{6} - 27844771778 p^{3} T^{7} + 282119759 p^{6} T^{8} - 1496510 p^{9} T^{9} + 24683 p^{12} T^{10} - 52 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( 1 - 340 T + 181850 T^{2} - 45373684 T^{3} + 13429117455 T^{4} - 2551173914240 T^{5} + 530229057162156 T^{6} - 2551173914240 p^{3} T^{7} + 13429117455 p^{6} T^{8} - 45373684 p^{9} T^{9} + 181850 p^{12} T^{10} - 340 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 - 332 T + 292071 T^{2} - 76574622 T^{3} + 36281439399 T^{4} - 7392542582574 T^{5} + 2436574283519458 T^{6} - 7392542582574 p^{3} T^{7} + 36281439399 p^{6} T^{8} - 76574622 p^{9} T^{9} + 292071 p^{12} T^{10} - 332 p^{15} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 - 616 T + 435339 T^{2} - 176922950 T^{3} + 74464996755 T^{4} - 22341033150810 T^{5} + 6778583702035218 T^{6} - 22341033150810 p^{3} T^{7} + 74464996755 p^{6} T^{8} - 176922950 p^{9} T^{9} + 435339 p^{12} T^{10} - 616 p^{15} T^{11} + p^{18} T^{12} \) | |
| 43 | \( 1 - 334 T + 430715 T^{2} - 123508878 T^{3} + 79745074707 T^{4} - 18904283647592 T^{5} + 8240498797811330 T^{6} - 18904283647592 p^{3} T^{7} + 79745074707 p^{6} T^{8} - 123508878 p^{9} T^{9} + 430715 p^{12} T^{10} - 334 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 - 85 T + 6534 p T^{2} - 27129259 T^{3} + 46333879375 T^{4} - 4270390938886 T^{5} + 5227445483301100 T^{6} - 4270390938886 p^{3} T^{7} + 46333879375 p^{6} T^{8} - 27129259 p^{9} T^{9} + 6534 p^{13} T^{10} - 85 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 - 850 T + 1116583 T^{2} - 652304118 T^{3} + 466050574283 T^{4} - 197142654656416 T^{5} + 95871162711193914 T^{6} - 197142654656416 p^{3} T^{7} + 466050574283 p^{6} T^{8} - 652304118 p^{9} T^{9} + 1116583 p^{12} T^{10} - 850 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 - 758 T + 1182306 T^{2} - 633358186 T^{3} + 565678561863 T^{4} - 231953762059260 T^{5} + 150395021848589820 T^{6} - 231953762059260 p^{3} T^{7} + 565678561863 p^{6} T^{8} - 633358186 p^{9} T^{9} + 1182306 p^{12} T^{10} - 758 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 + 36 T + 1049294 T^{2} + 61806916 T^{3} + 508396683767 T^{4} + 31304512521192 T^{5} + 145855929933536228 T^{6} + 31304512521192 p^{3} T^{7} + 508396683767 p^{6} T^{8} + 61806916 p^{9} T^{9} + 1049294 p^{12} T^{10} + 36 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 - 939 T + 1083468 T^{2} - 677838757 T^{3} + 553019898167 T^{4} - 325001005895942 T^{5} + 213102331026860744 T^{6} - 325001005895942 p^{3} T^{7} + 553019898167 p^{6} T^{8} - 677838757 p^{9} T^{9} + 1083468 p^{12} T^{10} - 939 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 - 1388 T + 1656622 T^{2} - 1864287268 T^{3} + 1492186295791 T^{4} - 1084857473044696 T^{5} + 736165848562113508 T^{6} - 1084857473044696 p^{3} T^{7} + 1492186295791 p^{6} T^{8} - 1864287268 p^{9} T^{9} + 1656622 p^{12} T^{10} - 1388 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 - 1708 T + 2388831 T^{2} - 2092640994 T^{3} + 1770401083215 T^{4} - 1181456884941858 T^{5} + 815155934532229474 T^{6} - 1181456884941858 p^{3} T^{7} + 1770401083215 p^{6} T^{8} - 2092640994 p^{9} T^{9} + 2388831 p^{12} T^{10} - 1708 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 - 1250 T + 2597586 T^{2} - 2405321470 T^{3} + 2913257383423 T^{4} - 2078370256058300 T^{5} + 1851933970653265820 T^{6} - 2078370256058300 p^{3} T^{7} + 2913257383423 p^{6} T^{8} - 2405321470 p^{9} T^{9} + 2597586 p^{12} T^{10} - 1250 p^{15} T^{11} + p^{18} T^{12} \) | |
| 83 | \( 1 - 748 T + 1762807 T^{2} - 1534149802 T^{3} + 1864138806963 T^{4} - 1377438064407974 T^{5} + 1341134023818679450 T^{6} - 1377438064407974 p^{3} T^{7} + 1864138806963 p^{6} T^{8} - 1534149802 p^{9} T^{9} + 1762807 p^{12} T^{10} - 748 p^{15} T^{11} + p^{18} T^{12} \) | |
| 89 | \( 1 + 1099 T + 3491726 T^{2} + 39764275 p T^{3} + 5580205971935 T^{4} + 4728799528650002 T^{5} + 5098038721965740676 T^{6} + 4728799528650002 p^{3} T^{7} + 5580205971935 p^{6} T^{8} + 39764275 p^{10} T^{9} + 3491726 p^{12} T^{10} + 1099 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 + 22 T + 2535151 T^{2} + 160798614 T^{3} + 3549398875803 T^{4} + 639134561324052 T^{5} + 3783183972148295146 T^{6} + 639134561324052 p^{3} T^{7} + 3549398875803 p^{6} T^{8} + 160798614 p^{9} T^{9} + 2535151 p^{12} T^{10} + 22 p^{15} T^{11} + p^{18} T^{12} \) | |
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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
−4.98370350037198139186335636093, −4.48836606616365249344251319079, −4.23982014287875410189527529179, −4.14133788314981326275958973523, −4.12863180188214425155256905578, −4.05559119200299172787473409784, −3.70167586759197876619678908477, −3.39574750201322704650397877823, −3.31521218844110823993166677023, −3.11957154074524617611821767225, −2.94054107922054638998877885450, −2.64022643579589217851844746362, −2.43289973869564801422534187091, −2.43089866597579298983439848808, −2.23978371928515366652093229612, −2.23571143552221040748030346376, −1.72574226942821449778547250817, −1.72167621806608542770626682447, −1.62451858496124770010980823076, −1.34101284007026192823134828260, −0.865463109453529623537002492798, −0.819810116994125049973339714016, −0.76591844627920522071778016409, −0.62430851322047967491468507308, −0.58070235365664815954997028375, 0.58070235365664815954997028375, 0.62430851322047967491468507308, 0.76591844627920522071778016409, 0.819810116994125049973339714016, 0.865463109453529623537002492798, 1.34101284007026192823134828260, 1.62451858496124770010980823076, 1.72167621806608542770626682447, 1.72574226942821449778547250817, 2.23571143552221040748030346376, 2.23978371928515366652093229612, 2.43089866597579298983439848808, 2.43289973869564801422534187091, 2.64022643579589217851844746362, 2.94054107922054638998877885450, 3.11957154074524617611821767225, 3.31521218844110823993166677023, 3.39574750201322704650397877823, 3.70167586759197876619678908477, 4.05559119200299172787473409784, 4.12863180188214425155256905578, 4.14133788314981326275958973523, 4.23982014287875410189527529179, 4.48836606616365249344251319079, 4.98370350037198139186335636093
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.