Dirichlet series
| L(s) = 1 | + 48·2-s − 162·3-s + 1.34e3·4-s + 203·5-s − 7.77e3·6-s − 2.05e3·7-s + 2.86e4·8-s + 1.53e4·9-s + 9.74e3·10-s − 2.69e3·11-s − 2.17e5·12-s + 1.31e4·13-s − 9.87e4·14-s − 3.28e4·15-s + 5.16e5·16-s + 2.91e3·17-s + 7.34e5·18-s − 1.30e4·19-s + 2.72e5·20-s + 3.33e5·21-s − 1.29e5·22-s + 1.15e4·23-s − 4.64e6·24-s − 1.79e5·25-s + 6.32e5·26-s − 1.10e6·27-s − 2.76e6·28-s + ⋯ |
| L(s) = 1 | + 4.24·2-s − 3.46·3-s + 21/2·4-s + 0.726·5-s − 14.6·6-s − 2.26·7-s + 19.7·8-s + 7·9-s + 3.08·10-s − 0.609·11-s − 36.3·12-s + 1.66·13-s − 9.62·14-s − 2.51·15-s + 63/2·16-s + 0.143·17-s + 29.6·18-s − 0.436·19-s + 7.62·20-s + 7.85·21-s − 2.58·22-s + 0.198·23-s − 68.5·24-s − 2.30·25-s + 7.06·26-s − 10.7·27-s − 23.8·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.46205\times 10^{13}\) |
| Root analytic conductor: | \(13.0599\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{6} ,\ ( \ : [7/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(81.78600400\) |
| \(L(\frac12)\) | \(\approx\) | \(81.78600400\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{3} T )^{6} \) |
| 3 | \( ( 1 + p^{3} T )^{6} \) | |
| 7 | \( ( 1 + p^{3} T )^{6} \) | |
| 13 | \( ( 1 - p^{3} T )^{6} \) | |
| good | 5 | \( 1 - 203 T + 220939 T^{2} - 61553648 T^{3} + 5838756941 p T^{4} - 299640767949 p^{2} T^{5} + 22025898037486 p^{3} T^{6} - 299640767949 p^{9} T^{7} + 5838756941 p^{15} T^{8} - 61553648 p^{21} T^{9} + 220939 p^{28} T^{10} - 203 p^{35} T^{11} + p^{42} T^{12} \) |
| 11 | \( 1 + 2690 T + 53515233 T^{2} + 190867649084 T^{3} + 1939765849757267 T^{4} + 6518069934289207106 T^{5} + \)\(38\!\cdots\!98\)\( p T^{6} + 6518069934289207106 p^{7} T^{7} + 1939765849757267 p^{14} T^{8} + 190867649084 p^{21} T^{9} + 53515233 p^{28} T^{10} + 2690 p^{35} T^{11} + p^{42} T^{12} \) | |
| 17 | \( 1 - 2910 T + 2025628953 T^{2} - 8105973662640 T^{3} + 1809736811967068439 T^{4} - \)\(75\!\cdots\!78\)\( T^{5} + \)\(94\!\cdots\!66\)\( T^{6} - \)\(75\!\cdots\!78\)\( p^{7} T^{7} + 1809736811967068439 p^{14} T^{8} - 8105973662640 p^{21} T^{9} + 2025628953 p^{28} T^{10} - 2910 p^{35} T^{11} + p^{42} T^{12} \) | |
| 19 | \( 1 + 13055 T + 3040276533 T^{2} + 37825582435106 T^{3} + 4313107518376328975 T^{4} + \)\(53\!\cdots\!75\)\( T^{5} + \)\(42\!\cdots\!50\)\( T^{6} + \)\(53\!\cdots\!75\)\( p^{7} T^{7} + 4313107518376328975 p^{14} T^{8} + 37825582435106 p^{21} T^{9} + 3040276533 p^{28} T^{10} + 13055 p^{35} T^{11} + p^{42} T^{12} \) | |
| 23 | \( 1 - 11581 T + 13237632101 T^{2} - 198625577042130 T^{3} + 85953608501003294847 T^{4} - \)\(12\!\cdots\!21\)\( T^{5} + \)\(35\!\cdots\!62\)\( T^{6} - \)\(12\!\cdots\!21\)\( p^{7} T^{7} + 85953608501003294847 p^{14} T^{8} - 198625577042130 p^{21} T^{9} + 13237632101 p^{28} T^{10} - 11581 p^{35} T^{11} + p^{42} T^{12} \) | |
| 29 | \( 1 + 92335 T - 2785871393 T^{2} - 840886093498164 T^{3} + \)\(32\!\cdots\!89\)\( T^{4} + \)\(36\!\cdots\!61\)\( T^{5} + \)\(21\!\cdots\!14\)\( T^{6} + \)\(36\!\cdots\!61\)\( p^{7} T^{7} + \)\(32\!\cdots\!89\)\( p^{14} T^{8} - 840886093498164 p^{21} T^{9} - 2785871393 p^{28} T^{10} + 92335 p^{35} T^{11} + p^{42} T^{12} \) | |
| 31 | \( 1 + 83081 T + 68391166206 T^{2} + 240107960854323 T^{3} + \)\(24\!\cdots\!75\)\( T^{4} - \)\(71\!\cdots\!74\)\( T^{5} + \)\(74\!\cdots\!76\)\( T^{6} - \)\(71\!\cdots\!74\)\( p^{7} T^{7} + \)\(24\!\cdots\!75\)\( p^{14} T^{8} + 240107960854323 p^{21} T^{9} + 68391166206 p^{28} T^{10} + 83081 p^{35} T^{11} + p^{42} T^{12} \) | |
| 37 | \( 1 + 265114 T + 364638167109 T^{2} + 83103185072554124 T^{3} + \)\(63\!\cdots\!39\)\( T^{4} + \)\(13\!\cdots\!30\)\( T^{5} + \)\(72\!\cdots\!70\)\( T^{6} + \)\(13\!\cdots\!30\)\( p^{7} T^{7} + \)\(63\!\cdots\!39\)\( p^{14} T^{8} + 83103185072554124 p^{21} T^{9} + 364638167109 p^{28} T^{10} + 265114 p^{35} T^{11} + p^{42} T^{12} \) | |
| 41 | \( 1 + 367468 T + 324858219622 T^{2} + 136876028830965324 T^{3} + \)\(94\!\cdots\!51\)\( T^{4} + \)\(36\!\cdots\!64\)\( T^{5} + \)\(16\!\cdots\!52\)\( T^{6} + \)\(36\!\cdots\!64\)\( p^{7} T^{7} + \)\(94\!\cdots\!51\)\( p^{14} T^{8} + 136876028830965324 p^{21} T^{9} + 324858219622 p^{28} T^{10} + 367468 p^{35} T^{11} + p^{42} T^{12} \) | |
| 43 | \( 1 - 454955 T + 579698680293 T^{2} + 19757277739507646 T^{3} + \)\(18\!\cdots\!07\)\( T^{4} - \)\(29\!\cdots\!91\)\( T^{5} + \)\(90\!\cdots\!58\)\( T^{6} - \)\(29\!\cdots\!91\)\( p^{7} T^{7} + \)\(18\!\cdots\!07\)\( p^{14} T^{8} + 19757277739507646 p^{21} T^{9} + 579698680293 p^{28} T^{10} - 454955 p^{35} T^{11} + p^{42} T^{12} \) | |
| 47 | \( 1 - 733973 T + 2356885761906 T^{2} - 1409515964836686935 T^{3} + \)\(25\!\cdots\!83\)\( T^{4} - \)\(12\!\cdots\!46\)\( T^{5} + \)\(16\!\cdots\!32\)\( T^{6} - \)\(12\!\cdots\!46\)\( p^{7} T^{7} + \)\(25\!\cdots\!83\)\( p^{14} T^{8} - 1409515964836686935 p^{21} T^{9} + 2356885761906 p^{28} T^{10} - 733973 p^{35} T^{11} + p^{42} T^{12} \) | |
| 53 | \( 1 + 1577379 T + 6314630905780 T^{2} + 8382461703090339467 T^{3} + \)\(17\!\cdots\!67\)\( T^{4} + \)\(18\!\cdots\!54\)\( T^{5} + \)\(26\!\cdots\!20\)\( T^{6} + \)\(18\!\cdots\!54\)\( p^{7} T^{7} + \)\(17\!\cdots\!67\)\( p^{14} T^{8} + 8382461703090339467 p^{21} T^{9} + 6314630905780 p^{28} T^{10} + 1577379 p^{35} T^{11} + p^{42} T^{12} \) | |
| 59 | \( 1 - 2062708 T + 12250946213362 T^{2} - 24600819488003261180 T^{3} + \)\(67\!\cdots\!27\)\( T^{4} - \)\(11\!\cdots\!44\)\( T^{5} + \)\(21\!\cdots\!24\)\( T^{6} - \)\(11\!\cdots\!44\)\( p^{7} T^{7} + \)\(67\!\cdots\!27\)\( p^{14} T^{8} - 24600819488003261180 p^{21} T^{9} + 12250946213362 p^{28} T^{10} - 2062708 p^{35} T^{11} + p^{42} T^{12} \) | |
| 61 | \( 1 + 271270 T + 15015294239327 T^{2} + 3538628174816965922 T^{3} + \)\(10\!\cdots\!47\)\( T^{4} + \)\(20\!\cdots\!88\)\( T^{5} + \)\(41\!\cdots\!30\)\( T^{6} + \)\(20\!\cdots\!88\)\( p^{7} T^{7} + \)\(10\!\cdots\!47\)\( p^{14} T^{8} + 3538628174816965922 p^{21} T^{9} + 15015294239327 p^{28} T^{10} + 271270 p^{35} T^{11} + p^{42} T^{12} \) | |
| 67 | \( 1 + 758674 T + 17220007450006 T^{2} + 25173185817390289430 T^{3} + \)\(18\!\cdots\!43\)\( T^{4} + \)\(23\!\cdots\!28\)\( T^{5} + \)\(14\!\cdots\!36\)\( T^{6} + \)\(23\!\cdots\!28\)\( p^{7} T^{7} + \)\(18\!\cdots\!43\)\( p^{14} T^{8} + 25173185817390289430 p^{21} T^{9} + 17220007450006 p^{28} T^{10} + 758674 p^{35} T^{11} + p^{42} T^{12} \) | |
| 71 | \( 1 + 6138216 T + 37638515591922 T^{2} + 1497283646516258744 p T^{3} + \)\(29\!\cdots\!75\)\( T^{4} + \)\(22\!\cdots\!48\)\( T^{5} + \)\(68\!\cdots\!80\)\( T^{6} + \)\(22\!\cdots\!48\)\( p^{7} T^{7} + \)\(29\!\cdots\!75\)\( p^{14} T^{8} + 1497283646516258744 p^{22} T^{9} + 37638515591922 p^{28} T^{10} + 6138216 p^{35} T^{11} + p^{42} T^{12} \) | |
| 73 | \( 1 - 6361979 T + 35197512527671 T^{2} - \)\(12\!\cdots\!12\)\( T^{3} + \)\(52\!\cdots\!29\)\( T^{4} - \)\(19\!\cdots\!89\)\( T^{5} + \)\(75\!\cdots\!86\)\( T^{6} - \)\(19\!\cdots\!89\)\( p^{7} T^{7} + \)\(52\!\cdots\!29\)\( p^{14} T^{8} - \)\(12\!\cdots\!12\)\( p^{21} T^{9} + 35197512527671 p^{28} T^{10} - 6361979 p^{35} T^{11} + p^{42} T^{12} \) | |
| 79 | \( 1 + 899781 T + 4461858490258 T^{2} - \)\(12\!\cdots\!05\)\( T^{3} - \)\(17\!\cdots\!25\)\( T^{4} + \)\(26\!\cdots\!18\)\( T^{5} + \)\(11\!\cdots\!48\)\( T^{6} + \)\(26\!\cdots\!18\)\( p^{7} T^{7} - \)\(17\!\cdots\!25\)\( p^{14} T^{8} - \)\(12\!\cdots\!05\)\( p^{21} T^{9} + 4461858490258 p^{28} T^{10} + 899781 p^{35} T^{11} + p^{42} T^{12} \) | |
| 83 | \( 1 - 3313561 T + 31014785270378 T^{2} - \)\(12\!\cdots\!71\)\( T^{3} + \)\(15\!\cdots\!15\)\( T^{4} - \)\(30\!\cdots\!02\)\( T^{5} + \)\(33\!\cdots\!92\)\( T^{6} - \)\(30\!\cdots\!02\)\( p^{7} T^{7} + \)\(15\!\cdots\!15\)\( p^{14} T^{8} - \)\(12\!\cdots\!71\)\( p^{21} T^{9} + 31014785270378 p^{28} T^{10} - 3313561 p^{35} T^{11} + p^{42} T^{12} \) | |
| 89 | \( 1 - 11210703 T + 165653945700864 T^{2} - \)\(12\!\cdots\!71\)\( T^{3} + \)\(12\!\cdots\!95\)\( T^{4} - \)\(88\!\cdots\!06\)\( T^{5} + \)\(73\!\cdots\!92\)\( T^{6} - \)\(88\!\cdots\!06\)\( p^{7} T^{7} + \)\(12\!\cdots\!95\)\( p^{14} T^{8} - \)\(12\!\cdots\!71\)\( p^{21} T^{9} + 165653945700864 p^{28} T^{10} - 11210703 p^{35} T^{11} + p^{42} T^{12} \) | |
| 97 | \( 1 - 28682643 T + 716169189655756 T^{2} - \)\(11\!\cdots\!27\)\( T^{3} + \)\(16\!\cdots\!79\)\( T^{4} - \)\(17\!\cdots\!22\)\( T^{5} + \)\(17\!\cdots\!92\)\( T^{6} - \)\(17\!\cdots\!22\)\( p^{7} T^{7} + \)\(16\!\cdots\!79\)\( p^{14} T^{8} - \)\(11\!\cdots\!27\)\( p^{21} T^{9} + 716169189655756 p^{28} T^{10} - 28682643 p^{35} T^{11} + p^{42} T^{12} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.94078292007021613939819867499, −4.26252800408652070030667469964, −4.19042417717757668110264450495, −4.08994524974066139496114122415, −4.04855794116770796142311025942, −4.02061072469092244201472916675, −4.00645866369269924869648568940, −3.30199301612934987951550330344, −3.23940837863509245379639811319, −3.21655153845135260281412088250, −3.10396132413206258965250578496, −3.04241986630704641352208129291, −2.73635179864832842677227804766, −2.05228556972014238253990509680, −1.97185202713545342121048899135, −1.96022026472936209138523627697, −1.81428550971953996067784098314, −1.78540108812861357021929440290, −1.78512072902961486989476527460, −0.911088770615909371501490906501, −0.810002406834040888644411851669, −0.76295838478442390323343570313, −0.61348723659052799218708976614, −0.47583157446015896603649076687, −0.22019230176235425505212316589, 0.22019230176235425505212316589, 0.47583157446015896603649076687, 0.61348723659052799218708976614, 0.76295838478442390323343570313, 0.810002406834040888644411851669, 0.911088770615909371501490906501, 1.78512072902961486989476527460, 1.78540108812861357021929440290, 1.81428550971953996067784098314, 1.96022026472936209138523627697, 1.97185202713545342121048899135, 2.05228556972014238253990509680, 2.73635179864832842677227804766, 3.04241986630704641352208129291, 3.10396132413206258965250578496, 3.21655153845135260281412088250, 3.23940837863509245379639811319, 3.30199301612934987951550330344, 4.00645866369269924869648568940, 4.02061072469092244201472916675, 4.04855794116770796142311025942, 4.08994524974066139496114122415, 4.19042417717757668110264450495, 4.26252800408652070030667469964, 4.94078292007021613939819867499
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.