Dirichlet series
| L(s) = 1 | − 24·2-s − 54·3-s + 336·4-s + 4·5-s + 1.29e3·6-s − 54·7-s − 3.58e3·8-s + 1.70e3·9-s − 96·10-s + 436·11-s − 1.81e4·12-s − 536·13-s + 1.29e3·14-s − 216·15-s + 3.22e4·16-s + 910·17-s − 4.08e4·18-s + 1.46e3·19-s + 1.34e3·20-s + 2.91e3·21-s − 1.04e4·22-s + 1.63e3·23-s + 1.93e5·24-s − 9.96e3·25-s + 1.28e4·26-s − 4.08e4·27-s − 1.81e4·28-s + ⋯ |
| L(s) = 1 | − 4.24·2-s − 3.46·3-s + 21/2·4-s + 0.0715·5-s + 14.6·6-s − 0.416·7-s − 19.7·8-s + 7·9-s − 0.303·10-s + 1.08·11-s − 36.3·12-s − 0.879·13-s + 1.76·14-s − 0.247·15-s + 63/2·16-s + 0.763·17-s − 29.6·18-s + 0.929·19-s + 0.751·20-s + 1.44·21-s − 4.60·22-s + 0.644·23-s + 68.5·24-s − 3.18·25-s + 3.73·26-s − 10.7·27-s − 4.37·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 59^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 59^{6}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 3^{6} \cdot 59^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.34952\times 10^{10}\) |
| Root analytic conductor: | \(7.53497\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{6} \cdot 3^{6} \cdot 59^{6} ,\ ( \ : [5/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(0.1717070144\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1717070144\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{2} T )^{6} \) |
| 3 | \( ( 1 + p^{2} T )^{6} \) | |
| 59 | \( ( 1 - p^{2} T )^{6} \) | |
| good | 5 | \( 1 - 4 T + 9976 T^{2} - 24948 p T^{3} + 53310484 T^{4} - 549060732 T^{5} + 202258672938 T^{6} - 549060732 p^{5} T^{7} + 53310484 p^{10} T^{8} - 24948 p^{16} T^{9} + 9976 p^{20} T^{10} - 4 p^{25} T^{11} + p^{30} T^{12} \) |
| 7 | \( 1 + 54 T + 68205 T^{2} + 4767660 T^{3} + 2293918392 T^{4} + 151309494990 T^{5} + 48220852126340 T^{6} + 151309494990 p^{5} T^{7} + 2293918392 p^{10} T^{8} + 4767660 p^{15} T^{9} + 68205 p^{20} T^{10} + 54 p^{25} T^{11} + p^{30} T^{12} \) | |
| 11 | \( 1 - 436 T + 459692 T^{2} - 183632200 T^{3} + 131733653636 T^{4} - 46356760425876 T^{5} + 25201169190214102 T^{6} - 46356760425876 p^{5} T^{7} + 131733653636 p^{10} T^{8} - 183632200 p^{15} T^{9} + 459692 p^{20} T^{10} - 436 p^{25} T^{11} + p^{30} T^{12} \) | |
| 13 | \( 1 + 536 T + 1465392 T^{2} + 532299940 T^{3} + 1031695721076 T^{4} + 308603989425296 T^{5} + 473973724798286186 T^{6} + 308603989425296 p^{5} T^{7} + 1031695721076 p^{10} T^{8} + 532299940 p^{15} T^{9} + 1465392 p^{20} T^{10} + 536 p^{25} T^{11} + p^{30} T^{12} \) | |
| 17 | \( 1 - 910 T + 5327533 T^{2} - 6875302608 T^{3} + 13602146644300 T^{4} - 19537318173990822 T^{5} + 1347867020225524932 p T^{6} - 19537318173990822 p^{5} T^{7} + 13602146644300 p^{10} T^{8} - 6875302608 p^{15} T^{9} + 5327533 p^{20} T^{10} - 910 p^{25} T^{11} + p^{30} T^{12} \) | |
| 19 | \( 1 - 1462 T + 8916041 T^{2} - 14160913148 T^{3} + 42997060527024 T^{4} - 58920857176633902 T^{5} + \)\(13\!\cdots\!56\)\( T^{6} - 58920857176633902 p^{5} T^{7} + 42997060527024 p^{10} T^{8} - 14160913148 p^{15} T^{9} + 8916041 p^{20} T^{10} - 1462 p^{25} T^{11} + p^{30} T^{12} \) | |
| 23 | \( 1 - 1634 T + 19993097 T^{2} - 11627861024 T^{3} + 141708184694840 T^{4} + 97826191108151802 T^{5} + \)\(70\!\cdots\!52\)\( T^{6} + 97826191108151802 p^{5} T^{7} + 141708184694840 p^{10} T^{8} - 11627861024 p^{15} T^{9} + 19993097 p^{20} T^{10} - 1634 p^{25} T^{11} + p^{30} T^{12} \) | |
| 29 | \( 1 + 1598 T + 47147293 T^{2} + 95521613928 T^{3} + 1504961484330256 T^{4} + 2588644547996915262 T^{5} + \)\(38\!\cdots\!28\)\( T^{6} + 2588644547996915262 p^{5} T^{7} + 1504961484330256 p^{10} T^{8} + 95521613928 p^{15} T^{9} + 47147293 p^{20} T^{10} + 1598 p^{25} T^{11} + p^{30} T^{12} \) | |
| 31 | \( 1 + 5670 T + 56784705 T^{2} + 11108963288 T^{3} + 1598512569578808 T^{4} + 5620816829310241218 T^{5} + \)\(96\!\cdots\!32\)\( T^{6} + 5620816829310241218 p^{5} T^{7} + 1598512569578808 p^{10} T^{8} + 11108963288 p^{15} T^{9} + 56784705 p^{20} T^{10} + 5670 p^{25} T^{11} + p^{30} T^{12} \) | |
| 37 | \( 1 + 20458 T + 464621453 T^{2} + 6288176042120 T^{3} + 83743654912254456 T^{4} + \)\(82\!\cdots\!34\)\( T^{5} + \)\(78\!\cdots\!32\)\( T^{6} + \)\(82\!\cdots\!34\)\( p^{5} T^{7} + 83743654912254456 p^{10} T^{8} + 6288176042120 p^{15} T^{9} + 464621453 p^{20} T^{10} + 20458 p^{25} T^{11} + p^{30} T^{12} \) | |
| 41 | \( 1 - 262 T + 223170001 T^{2} + 1390474910376 T^{3} + 33821786408572684 T^{4} + \)\(30\!\cdots\!46\)\( T^{5} + \)\(40\!\cdots\!32\)\( T^{6} + \)\(30\!\cdots\!46\)\( p^{5} T^{7} + 33821786408572684 p^{10} T^{8} + 1390474910376 p^{15} T^{9} + 223170001 p^{20} T^{10} - 262 p^{25} T^{11} + p^{30} T^{12} \) | |
| 43 | \( 1 + 34028 T + 915077400 T^{2} + 17236461278512 T^{3} + 304953523516603248 T^{4} + \)\(43\!\cdots\!76\)\( T^{5} + \)\(58\!\cdots\!90\)\( T^{6} + \)\(43\!\cdots\!76\)\( p^{5} T^{7} + 304953523516603248 p^{10} T^{8} + 17236461278512 p^{15} T^{9} + 915077400 p^{20} T^{10} + 34028 p^{25} T^{11} + p^{30} T^{12} \) | |
| 47 | \( 1 + 11194 T + 672469985 T^{2} + 5594346229564 T^{3} + 255795000411238112 T^{4} + \)\(16\!\cdots\!54\)\( T^{5} + \)\(66\!\cdots\!08\)\( T^{6} + \)\(16\!\cdots\!54\)\( p^{5} T^{7} + 255795000411238112 p^{10} T^{8} + 5594346229564 p^{15} T^{9} + 672469985 p^{20} T^{10} + 11194 p^{25} T^{11} + p^{30} T^{12} \) | |
| 53 | \( 1 + 17164 T + 1677524616 T^{2} + 25281081254380 T^{3} + 1408286807531936388 T^{4} + \)\(18\!\cdots\!20\)\( T^{5} + \)\(72\!\cdots\!78\)\( T^{6} + \)\(18\!\cdots\!20\)\( p^{5} T^{7} + 1408286807531936388 p^{10} T^{8} + 25281081254380 p^{15} T^{9} + 1677524616 p^{20} T^{10} + 17164 p^{25} T^{11} + p^{30} T^{12} \) | |
| 61 | \( 1 + 43546 T + 5385255733 T^{2} + 177426474083956 T^{3} + 11709138441318894480 T^{4} + \)\(29\!\cdots\!18\)\( T^{5} + \)\(13\!\cdots\!20\)\( T^{6} + \)\(29\!\cdots\!18\)\( p^{5} T^{7} + 11709138441318894480 p^{10} T^{8} + 177426474083956 p^{15} T^{9} + 5385255733 p^{20} T^{10} + 43546 p^{25} T^{11} + p^{30} T^{12} \) | |
| 67 | \( 1 + 52772 T + 3382954736 T^{2} + 158662858801504 T^{3} + 9478623811913430408 T^{4} + \)\(32\!\cdots\!76\)\( T^{5} + \)\(13\!\cdots\!14\)\( T^{6} + \)\(32\!\cdots\!76\)\( p^{5} T^{7} + 9478623811913430408 p^{10} T^{8} + 158662858801504 p^{15} T^{9} + 3382954736 p^{20} T^{10} + 52772 p^{25} T^{11} + p^{30} T^{12} \) | |
| 71 | \( 1 - 84740 T + 11459150268 T^{2} - 699721220301656 T^{3} + 53123891526085177560 T^{4} - \)\(24\!\cdots\!12\)\( T^{5} + \)\(12\!\cdots\!94\)\( T^{6} - \)\(24\!\cdots\!12\)\( p^{5} T^{7} + 53123891526085177560 p^{10} T^{8} - 699721220301656 p^{15} T^{9} + 11459150268 p^{20} T^{10} - 84740 p^{25} T^{11} + p^{30} T^{12} \) | |
| 73 | \( 1 + 36578 T + 3518167769 T^{2} - 27668354351060 T^{3} + 7382227026611138340 T^{4} + \)\(12\!\cdots\!74\)\( T^{5} + \)\(31\!\cdots\!68\)\( T^{6} + \)\(12\!\cdots\!74\)\( p^{5} T^{7} + 7382227026611138340 p^{10} T^{8} - 27668354351060 p^{15} T^{9} + 3518167769 p^{20} T^{10} + 36578 p^{25} T^{11} + p^{30} T^{12} \) | |
| 79 | \( 1 - 85196 T + 15926008556 T^{2} - 961169876566552 T^{3} + \)\(10\!\cdots\!32\)\( T^{4} - \)\(51\!\cdots\!24\)\( T^{5} + \)\(42\!\cdots\!50\)\( T^{6} - \)\(51\!\cdots\!24\)\( p^{5} T^{7} + \)\(10\!\cdots\!32\)\( p^{10} T^{8} - 961169876566552 p^{15} T^{9} + 15926008556 p^{20} T^{10} - 85196 p^{25} T^{11} + p^{30} T^{12} \) | |
| 83 | \( 1 - 217026 T + 31103999493 T^{2} - 3323236881536672 T^{3} + \)\(29\!\cdots\!72\)\( T^{4} - \)\(21\!\cdots\!50\)\( T^{5} + \)\(14\!\cdots\!96\)\( T^{6} - \)\(21\!\cdots\!50\)\( p^{5} T^{7} + \)\(29\!\cdots\!72\)\( p^{10} T^{8} - 3323236881536672 p^{15} T^{9} + 31103999493 p^{20} T^{10} - 217026 p^{25} T^{11} + p^{30} T^{12} \) | |
| 89 | \( 1 - 333850 T + 69762239049 T^{2} - 10651861736275516 T^{3} + \)\(12\!\cdots\!84\)\( T^{4} - \)\(12\!\cdots\!30\)\( T^{5} + \)\(10\!\cdots\!80\)\( T^{6} - \)\(12\!\cdots\!30\)\( p^{5} T^{7} + \)\(12\!\cdots\!84\)\( p^{10} T^{8} - 10651861736275516 p^{15} T^{9} + 69762239049 p^{20} T^{10} - 333850 p^{25} T^{11} + p^{30} T^{12} \) | |
| 97 | \( 1 - 173148 T + 25314290508 T^{2} - 2373696444790708 T^{3} + \)\(22\!\cdots\!64\)\( T^{4} - \)\(17\!\cdots\!04\)\( T^{5} + \)\(18\!\cdots\!54\)\( T^{6} - \)\(17\!\cdots\!04\)\( p^{5} T^{7} + \)\(22\!\cdots\!64\)\( p^{10} T^{8} - 2373696444790708 p^{15} T^{9} + 25314290508 p^{20} T^{10} - 173148 p^{25} T^{11} + p^{30} T^{12} \) | |
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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
−5.63435502996892920824172290168, −5.07789283855646793077848986318, −5.04237990477402460140903429968, −5.01475495789738130735331814509, −4.72487596350321619461162690342, −4.63684498954185625589933704043, −4.53024788618163106151372581446, −3.65406285526802359757148546293, −3.57490857694018078936123309550, −3.42035141572560232761162566658, −3.37779725371944122864011151813, −3.28729048793884930325793374234, −3.14141780603159005045269050243, −2.09914135990295325139130465148, −2.03084294299612777385365779972, −1.89286594632566572776291648931, −1.88482348598650722023035153655, −1.72450746605498179717825897961, −1.69616157155046578130593369696, −1.02816397757589297201835643619, −0.799395774092365461821550347823, −0.62562173491438070356190351986, −0.51921619788166944104351660330, −0.41120633170087273669928793384, −0.23179998803958650912217498006, 0.23179998803958650912217498006, 0.41120633170087273669928793384, 0.51921619788166944104351660330, 0.62562173491438070356190351986, 0.799395774092365461821550347823, 1.02816397757589297201835643619, 1.69616157155046578130593369696, 1.72450746605498179717825897961, 1.88482348598650722023035153655, 1.89286594632566572776291648931, 2.03084294299612777385365779972, 2.09914135990295325139130465148, 3.14141780603159005045269050243, 3.28729048793884930325793374234, 3.37779725371944122864011151813, 3.42035141572560232761162566658, 3.57490857694018078936123309550, 3.65406285526802359757148546293, 4.53024788618163106151372581446, 4.63684498954185625589933704043, 4.72487596350321619461162690342, 5.01475495789738130735331814509, 5.04237990477402460140903429968, 5.07789283855646793077848986318, 5.63435502996892920824172290168
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.