Dirichlet series
| L(s) = 1 | + 2·3-s + 12·5-s + 2·9-s + 24·15-s + 180·19-s − 120·23-s − 528·25-s − 42·27-s + 588·29-s − 372·43-s + 24·45-s + 1.24e3·47-s + 1.58e3·49-s − 948·53-s + 360·57-s + 2.29e3·67-s − 240·69-s − 2.04e3·71-s + 216·73-s − 1.05e3·75-s − 355·81-s + 1.17e3·87-s + 2.16e3·95-s − 48·97-s − 4.93e3·101-s − 1.44e3·115-s + 7.34e3·121-s + ⋯ |
| L(s) = 1 | + 0.384·3-s + 1.07·5-s + 2/27·9-s + 0.413·15-s + 2.17·19-s − 1.08·23-s − 4.22·25-s − 0.299·27-s + 3.76·29-s − 1.31·43-s + 0.0795·45-s + 3.87·47-s + 4.61·49-s − 2.45·53-s + 0.836·57-s + 4.17·67-s − 0.418·69-s − 3.40·71-s + 0.346·73-s − 1.62·75-s − 0.486·81-s + 1.44·87-s + 2.33·95-s − 0.0502·97-s − 4.85·101-s − 1.16·115-s + 5.51·121-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{96} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{96} \cdot 3^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.49423\times 10^{19}\) |
| Root analytic conductor: | \(6.73152\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{96} \cdot 3^{12} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(53.79200040\) |
| \(L(\frac12)\) | \(\approx\) | \(53.79200040\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 2 T + 2 T^{2} + 14 p T^{3} + 61 p T^{4} + 388 p^{2} T^{5} - 1060 p^{2} T^{6} + 388 p^{5} T^{7} + 61 p^{7} T^{8} + 14 p^{10} T^{9} + 2 p^{12} T^{10} - 2 p^{15} T^{11} + p^{18} T^{12} \) | |
| good | 5 | \( ( 1 - 6 T + 318 T^{2} - 142 p^{2} T^{3} + 62343 T^{4} - 836484 T^{5} + 8360964 T^{6} - 836484 p^{3} T^{7} + 62343 p^{6} T^{8} - 142 p^{11} T^{9} + 318 p^{12} T^{10} - 6 p^{15} T^{11} + p^{18} T^{12} )^{2} \) |
| 7 | \( 1 - 1584 T^{2} + 1166442 T^{4} - 478563952 T^{6} + 86370897327 T^{8} + 18087151507872 T^{10} - 14510145747466484 T^{12} + 18087151507872 p^{6} T^{14} + 86370897327 p^{12} T^{16} - 478563952 p^{18} T^{18} + 1166442 p^{24} T^{20} - 1584 p^{30} T^{22} + p^{36} T^{24} \) | |
| 11 | \( 1 - 7344 T^{2} + 27856218 T^{4} - 72819366256 T^{6} + 147572972668479 T^{8} - 246996158362631520 T^{10} + \)\(35\!\cdots\!40\)\( T^{12} - 246996158362631520 p^{6} T^{14} + 147572972668479 p^{12} T^{16} - 72819366256 p^{18} T^{18} + 27856218 p^{24} T^{20} - 7344 p^{30} T^{22} + p^{36} T^{24} \) | |
| 13 | \( 1 - 11460 T^{2} + 71492658 T^{4} - 321614164500 T^{6} + 1140360023231295 T^{8} - 3295200150108212232 T^{10} + 46796507939684390300 p^{2} T^{12} - 3295200150108212232 p^{6} T^{14} + 1140360023231295 p^{12} T^{16} - 321614164500 p^{18} T^{18} + 71492658 p^{24} T^{20} - 11460 p^{30} T^{22} + p^{36} T^{24} \) | |
| 17 | \( 1 - 31164 T^{2} + 484190562 T^{4} - 4991368612172 T^{6} + 38572842772586415 T^{8} - \)\(24\!\cdots\!92\)\( T^{10} + \)\(12\!\cdots\!56\)\( T^{12} - \)\(24\!\cdots\!92\)\( p^{6} T^{14} + 38572842772586415 p^{12} T^{16} - 4991368612172 p^{18} T^{18} + 484190562 p^{24} T^{20} - 31164 p^{30} T^{22} + p^{36} T^{24} \) | |
| 19 | \( ( 1 - 90 T + 23130 T^{2} - 1130334 T^{3} + 238485783 T^{4} - 8954726028 T^{5} + 1886810487948 T^{6} - 8954726028 p^{3} T^{7} + 238485783 p^{6} T^{8} - 1130334 p^{9} T^{9} + 23130 p^{12} T^{10} - 90 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 23 | \( ( 1 + 60 T + 35946 T^{2} + 1305844 T^{3} + 618033375 T^{4} + 10828626072 T^{5} + 7737521174604 T^{6} + 10828626072 p^{3} T^{7} + 618033375 p^{6} T^{8} + 1305844 p^{9} T^{9} + 35946 p^{12} T^{10} + 60 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 29 | \( ( 1 - 294 T + 130110 T^{2} - 26159150 T^{3} + 7028600823 T^{4} - 1115002393092 T^{5} + 221541290194500 T^{6} - 1115002393092 p^{3} T^{7} + 7028600823 p^{6} T^{8} - 26159150 p^{9} T^{9} + 130110 p^{12} T^{10} - 294 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 31 | \( 1 - 188448 T^{2} + 548969046 p T^{4} - 999868768809888 T^{6} + 44197970839051250895 T^{8} - \)\(16\!\cdots\!56\)\( T^{10} + \)\(51\!\cdots\!24\)\( T^{12} - \)\(16\!\cdots\!56\)\( p^{6} T^{14} + 44197970839051250895 p^{12} T^{16} - 999868768809888 p^{18} T^{18} + 548969046 p^{25} T^{20} - 188448 p^{30} T^{22} + p^{36} T^{24} \) | |
| 37 | \( 1 - 289860 T^{2} + 39162595986 T^{4} - 3448567340426452 T^{6} + \)\(24\!\cdots\!39\)\( T^{8} - \)\(14\!\cdots\!84\)\( T^{10} + \)\(80\!\cdots\!44\)\( T^{12} - \)\(14\!\cdots\!84\)\( p^{6} T^{14} + \)\(24\!\cdots\!39\)\( p^{12} T^{16} - 3448567340426452 p^{18} T^{18} + 39162595986 p^{24} T^{20} - 289860 p^{30} T^{22} + p^{36} T^{24} \) | |
| 41 | \( 1 - 472860 T^{2} + 113880883458 T^{4} - 18423203395045100 T^{6} + \)\(22\!\cdots\!39\)\( T^{8} - \)\(21\!\cdots\!48\)\( T^{10} + \)\(16\!\cdots\!96\)\( T^{12} - \)\(21\!\cdots\!48\)\( p^{6} T^{14} + \)\(22\!\cdots\!39\)\( p^{12} T^{16} - 18423203395045100 p^{18} T^{18} + 113880883458 p^{24} T^{20} - 472860 p^{30} T^{22} + p^{36} T^{24} \) | |
| 43 | \( ( 1 + 186 T + 210426 T^{2} + 28021582 T^{3} + 21001670439 T^{4} + 1480704807564 T^{5} + 1626060540071308 T^{6} + 1480704807564 p^{3} T^{7} + 21001670439 p^{6} T^{8} + 28021582 p^{9} T^{9} + 210426 p^{12} T^{10} + 186 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 47 | \( ( 1 - 624 T + 9222 p T^{2} - 110814928 T^{3} + 31731910959 T^{4} + 3786299059104 T^{5} - 661034171971540 T^{6} + 3786299059104 p^{3} T^{7} + 31731910959 p^{6} T^{8} - 110814928 p^{9} T^{9} + 9222 p^{13} T^{10} - 624 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 53 | \( ( 1 + 474 T + 500382 T^{2} + 163987106 T^{3} + 125315372775 T^{4} + 36437344230588 T^{5} + 23049801332671428 T^{6} + 36437344230588 p^{3} T^{7} + 125315372775 p^{6} T^{8} + 163987106 p^{9} T^{9} + 500382 p^{12} T^{10} + 474 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 59 | \( 1 - 664992 T^{2} + 220351665594 T^{4} - 57825701639377952 T^{6} + \)\(16\!\cdots\!91\)\( T^{8} - \)\(40\!\cdots\!28\)\( T^{10} + \)\(89\!\cdots\!68\)\( T^{12} - \)\(40\!\cdots\!28\)\( p^{6} T^{14} + \)\(16\!\cdots\!91\)\( p^{12} T^{16} - 57825701639377952 p^{18} T^{18} + 220351665594 p^{24} T^{20} - 664992 p^{30} T^{22} + p^{36} T^{24} \) | |
| 61 | \( 1 - 920868 T^{2} + 521744273202 T^{4} - 210274590611898484 T^{6} + \)\(68\!\cdots\!63\)\( T^{8} - \)\(18\!\cdots\!12\)\( T^{10} + \)\(45\!\cdots\!52\)\( T^{12} - \)\(18\!\cdots\!12\)\( p^{6} T^{14} + \)\(68\!\cdots\!63\)\( p^{12} T^{16} - 210274590611898484 p^{18} T^{18} + 521744273202 p^{24} T^{20} - 920868 p^{30} T^{22} + p^{36} T^{24} \) | |
| 67 | \( ( 1 - 1146 T + 1800066 T^{2} - 1517084862 T^{3} + 1324387371063 T^{4} - 851199621272364 T^{5} + 525406522359528380 T^{6} - 851199621272364 p^{3} T^{7} + 1324387371063 p^{6} T^{8} - 1517084862 p^{9} T^{9} + 1800066 p^{12} T^{10} - 1146 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 71 | \( ( 1 + 1020 T + 2040810 T^{2} + 1572918772 T^{3} + 1765475696511 T^{4} + 1048763672220696 T^{5} + 832227063915098828 T^{6} + 1048763672220696 p^{3} T^{7} + 1765475696511 p^{6} T^{8} + 1572918772 p^{9} T^{9} + 2040810 p^{12} T^{10} + 1020 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 73 | \( ( 1 - 108 T + 839826 T^{2} - 159624796 T^{3} + 499086952191 T^{4} - 64393086787416 T^{5} + 241123492212399420 T^{6} - 64393086787416 p^{3} T^{7} + 499086952191 p^{6} T^{8} - 159624796 p^{9} T^{9} + 839826 p^{12} T^{10} - 108 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 79 | \( 1 - 4787328 T^{2} + 10929285260394 T^{4} - 15723000113198264960 T^{6} + \)\(15\!\cdots\!59\)\( T^{8} - \)\(11\!\cdots\!88\)\( T^{10} + \)\(66\!\cdots\!56\)\( T^{12} - \)\(11\!\cdots\!88\)\( p^{6} T^{14} + \)\(15\!\cdots\!59\)\( p^{12} T^{16} - 15723000113198264960 p^{18} T^{18} + 10929285260394 p^{24} T^{20} - 4787328 p^{30} T^{22} + p^{36} T^{24} \) | |
| 83 | \( 1 - 4105392 T^{2} + 7811084838138 T^{4} - 9357451093879929968 T^{6} + \)\(81\!\cdots\!15\)\( T^{8} - \)\(58\!\cdots\!92\)\( T^{10} + \)\(35\!\cdots\!24\)\( T^{12} - \)\(58\!\cdots\!92\)\( p^{6} T^{14} + \)\(81\!\cdots\!15\)\( p^{12} T^{16} - 9357451093879929968 p^{18} T^{18} + 7811084838138 p^{24} T^{20} - 4105392 p^{30} T^{22} + p^{36} T^{24} \) | |
| 89 | \( 1 - 4768524 T^{2} + 12015644253474 T^{4} - 230672060265750012 p T^{6} + \)\(26\!\cdots\!15\)\( T^{8} - \)\(25\!\cdots\!96\)\( T^{10} + \)\(20\!\cdots\!60\)\( T^{12} - \)\(25\!\cdots\!96\)\( p^{6} T^{14} + \)\(26\!\cdots\!15\)\( p^{12} T^{16} - 230672060265750012 p^{19} T^{18} + 12015644253474 p^{24} T^{20} - 4768524 p^{30} T^{22} + p^{36} T^{24} \) | |
| 97 | \( ( 1 + 24 T + 3053178 T^{2} + 1259266232 T^{3} + 4503411377103 T^{4} + 2753933866197552 T^{5} + 48632588768677228 p T^{6} + 2753933866197552 p^{3} T^{7} + 4503411377103 p^{6} T^{8} + 1259266232 p^{9} T^{9} + 3053178 p^{12} T^{10} + 24 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.85091596234192445406053172474, −2.82341081086041772178661773703, −2.82241187305665831788570102741, −2.78986481880379078772846101517, −2.60142226058834307656099961869, −2.17358645883827702028693090653, −2.12284152873379567378987938816, −2.09236621115103779592076525307, −1.98420245944748182024804946065, −1.90992007401793941483229833407, −1.86664423341931714760654753508, −1.85039516088338196760032437120, −1.74172877821587152423117613732, −1.56474341827714296570329062055, −1.50557506485881272274930772177, −1.16743186637614268780271655993, −1.03058394669958694087015713055, −0.880757089776277336926198366418, −0.856499148492855464390948390948, −0.828419743367840238506586819020, −0.61730864749633121338514714538, −0.57113392962517758285710394361, −0.43229609011323894403651744478, −0.31515247308560671468474320076, −0.15467224628908288893633255758, 0.15467224628908288893633255758, 0.31515247308560671468474320076, 0.43229609011323894403651744478, 0.57113392962517758285710394361, 0.61730864749633121338514714538, 0.828419743367840238506586819020, 0.856499148492855464390948390948, 0.880757089776277336926198366418, 1.03058394669958694087015713055, 1.16743186637614268780271655993, 1.50557506485881272274930772177, 1.56474341827714296570329062055, 1.74172877821587152423117613732, 1.85039516088338196760032437120, 1.86664423341931714760654753508, 1.90992007401793941483229833407, 1.98420245944748182024804946065, 2.09236621115103779592076525307, 2.12284152873379567378987938816, 2.17358645883827702028693090653, 2.60142226058834307656099961869, 2.78986481880379078772846101517, 2.82241187305665831788570102741, 2.82341081086041772178661773703, 2.85091596234192445406053172474
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.