Dirichlet series
| L(s) = 1 | + 233·3-s − 733·5-s − 5.01e3·7-s + 4.91e4·9-s − 7.33e3·11-s + 1.97e5·13-s − 1.70e5·15-s − 3.06e5·17-s + 3.77e5·19-s − 1.16e6·21-s + 2.26e6·23-s + 3.12e6·25-s + 3.17e6·27-s − 1.30e7·29-s + 6.65e6·31-s − 1.70e6·33-s + 3.67e6·35-s − 2.22e7·37-s + 4.59e7·39-s + 6.80e7·41-s + 1.25e8·43-s − 3.60e7·45-s + 5.27e7·47-s + 4.13e7·49-s − 7.14e7·51-s − 1.20e7·53-s + 5.37e6·55-s + ⋯ |
| L(s) = 1 | + 1.66·3-s − 0.524·5-s − 0.788·7-s + 2.49·9-s − 0.151·11-s + 1.91·13-s − 0.871·15-s − 0.890·17-s + 0.665·19-s − 1.31·21-s + 1.68·23-s + 1.60·25-s + 1.14·27-s − 3.43·29-s + 1.29·31-s − 0.251·33-s + 0.413·35-s − 1.95·37-s + 3.17·39-s + 3.76·41-s + 5.60·43-s − 1.30·45-s + 1.57·47-s + 1.02·49-s − 1.47·51-s − 0.210·53-s + 0.0792·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{24} \cdot 7^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.68411\times 10^{10}\) |
| Root analytic conductor: | \(7.59499\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{24} \cdot 7^{6} ,\ ( \ : [9/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(15.59058742\) |
| \(L(\frac12)\) | \(\approx\) | \(15.59058742\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 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 | 3 | \( 1 - 233 T + 5149 T^{2} + 786158 p^{2} T^{3} - 78961157 p^{2} T^{4} - 389726399 p^{5} T^{5} + 401030861374 p^{4} T^{6} - 389726399 p^{14} T^{7} - 78961157 p^{20} T^{8} + 786158 p^{29} T^{9} + 5149 p^{36} T^{10} - 233 p^{45} T^{11} + p^{54} T^{12} \) |
| 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
−5.79269611499724634209634579632, −5.79062986402482790012473331673, −5.62089406241809985595787483084, −5.15772521116385414519297978378, −4.70380732397408470933896553083, −4.45768466992005862685242110796, −4.31671717641658742217640705166, −4.22665674854952222474818469175, −4.00611563073335929938596670461, −3.73143600233812282524124567387, −3.63258770019610523677960048234, −3.43001663674976718581226579243, −3.19676563716586737118920754537, −2.63918069325227259174741978121, −2.60317494982930376849903997559, −2.54218245694974252286187518227, −2.47599342828489948513873965213, −1.85138989928159099254302703264, −1.76151803422820221911403792883, −1.19577800606378995487195890702, −1.17575387596188296463319455770, −1.13687397869607575557061428364, −0.64076381775357206044423029166, −0.58326112921567431512309324550, −0.20449819085918260279422807843, 0.20449819085918260279422807843, 0.58326112921567431512309324550, 0.64076381775357206044423029166, 1.13687397869607575557061428364, 1.17575387596188296463319455770, 1.19577800606378995487195890702, 1.76151803422820221911403792883, 1.85138989928159099254302703264, 2.47599342828489948513873965213, 2.54218245694974252286187518227, 2.60317494982930376849903997559, 2.63918069325227259174741978121, 3.19676563716586737118920754537, 3.43001663674976718581226579243, 3.63258770019610523677960048234, 3.73143600233812282524124567387, 4.00611563073335929938596670461, 4.22665674854952222474818469175, 4.31671717641658742217640705166, 4.45768466992005862685242110796, 4.70380732397408470933896553083, 5.15772521116385414519297978378, 5.62089406241809985595787483084, 5.79062986402482790012473331673, 5.79269611499724634209634579632
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.