Dirichlet series
| L(s) = 1 | + 5·3-s − 5·5-s + 38·7-s − 45·9-s + 19·11-s + 13·13-s − 25·15-s − 218·17-s + 290·19-s + 190·21-s + 196·23-s − 369·25-s − 166·27-s − 174·29-s + 675·31-s + 95·33-s − 190·35-s − 238·37-s + 65·39-s − 464·41-s + 579·43-s + 225·45-s + 975·47-s + 150·49-s − 1.09e3·51-s + 515·53-s − 95·55-s + ⋯ |
| L(s) = 1 | + 0.962·3-s − 0.447·5-s + 2.05·7-s − 5/3·9-s + 0.520·11-s + 0.277·13-s − 0.430·15-s − 3.11·17-s + 3.50·19-s + 1.97·21-s + 1.77·23-s − 2.95·25-s − 1.18·27-s − 1.11·29-s + 3.91·31-s + 0.501·33-s − 0.917·35-s − 1.05·37-s + 0.266·39-s − 1.76·41-s + 2.05·43-s + 0.745·45-s + 3.02·47-s + 0.437·49-s − 2.99·51-s + 1.33·53-s − 0.232·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{24} \cdot 29^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.21021\times 10^{8}\) |
| Root analytic conductor: | \(5.23229\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{24} \cdot 29^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(17.00494822\) |
| \(L(\frac12)\) | \(\approx\) | \(17.00494822\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 29 | \( ( 1 + p T )^{6} \) | |
| good | 3 | \( 1 - 5 T + 70 T^{2} - 409 T^{3} + 860 p T^{4} - 4751 p T^{5} + 76528 T^{6} - 4751 p^{4} T^{7} + 860 p^{7} T^{8} - 409 p^{9} T^{9} + 70 p^{12} T^{10} - 5 p^{15} T^{11} + p^{18} T^{12} \) |
| 5 | \( 1 + p T + 394 T^{2} + 1787 T^{3} + 85964 T^{4} + 307713 T^{5} + 12844828 T^{6} + 307713 p^{3} T^{7} + 85964 p^{6} T^{8} + 1787 p^{9} T^{9} + 394 p^{12} T^{10} + p^{16} T^{11} + p^{18} T^{12} \) | |
| 7 | \( 1 - 38 T + 1294 T^{2} - 4278 p T^{3} + 583567 T^{4} - 9312452 T^{5} + 176544740 T^{6} - 9312452 p^{3} T^{7} + 583567 p^{6} T^{8} - 4278 p^{10} T^{9} + 1294 p^{12} T^{10} - 38 p^{15} T^{11} + p^{18} T^{12} \) | |
| 11 | \( 1 - 19 T + 2642 T^{2} - 65039 T^{3} + 5795748 T^{4} - 142815547 T^{5} + 7955435996 T^{6} - 142815547 p^{3} T^{7} + 5795748 p^{6} T^{8} - 65039 p^{9} T^{9} + 2642 p^{12} T^{10} - 19 p^{15} T^{11} + p^{18} T^{12} \) | |
| 13 | \( 1 - p T + 4842 T^{2} - 138651 T^{3} + 15279148 T^{4} - 598048473 T^{5} + 33610235148 T^{6} - 598048473 p^{3} T^{7} + 15279148 p^{6} T^{8} - 138651 p^{9} T^{9} + 4842 p^{12} T^{10} - p^{16} T^{11} + p^{18} T^{12} \) | |
| 17 | \( 1 + 218 T + 24866 T^{2} + 1410330 T^{3} + 16045023 T^{4} - 6011814932 T^{5} - 605183485764 T^{6} - 6011814932 p^{3} T^{7} + 16045023 p^{6} T^{8} + 1410330 p^{9} T^{9} + 24866 p^{12} T^{10} + 218 p^{15} T^{11} + p^{18} T^{12} \) | |
| 19 | \( 1 - 290 T + 44866 T^{2} - 5140110 T^{3} + 545630615 T^{4} - 56569682804 T^{5} + 5165167815164 T^{6} - 56569682804 p^{3} T^{7} + 545630615 p^{6} T^{8} - 5140110 p^{9} T^{9} + 44866 p^{12} T^{10} - 290 p^{15} T^{11} + p^{18} T^{12} \) | |
| 23 | \( 1 - 196 T + 1346 p T^{2} - 1229644 T^{3} + 63545295 T^{4} + 14534767960 T^{5} + 2257525436 p T^{6} + 14534767960 p^{3} T^{7} + 63545295 p^{6} T^{8} - 1229644 p^{9} T^{9} + 1346 p^{13} T^{10} - 196 p^{15} T^{11} + p^{18} T^{12} \) | |
| 31 | \( 1 - 675 T + 321070 T^{2} - 108047311 T^{3} + 29745345672 T^{4} - 6638068285827 T^{5} + 1256031688770516 T^{6} - 6638068285827 p^{3} T^{7} + 29745345672 p^{6} T^{8} - 108047311 p^{9} T^{9} + 321070 p^{12} T^{10} - 675 p^{15} T^{11} + p^{18} T^{12} \) | |
| 37 | \( 1 + 238 T + 200510 T^{2} + 45340750 T^{3} + 21090986759 T^{4} + 4050456264108 T^{5} + 1329213380793796 T^{6} + 4050456264108 p^{3} T^{7} + 21090986759 p^{6} T^{8} + 45340750 p^{9} T^{9} + 200510 p^{12} T^{10} + 238 p^{15} T^{11} + p^{18} T^{12} \) | |
| 41 | \( 1 + 464 T + 317198 T^{2} + 101440240 T^{3} + 42424737871 T^{4} + 10869150873536 T^{5} + 3553602559791268 T^{6} + 10869150873536 p^{3} T^{7} + 42424737871 p^{6} T^{8} + 101440240 p^{9} T^{9} + 317198 p^{12} T^{10} + 464 p^{15} T^{11} + p^{18} T^{12} \) | |
| 43 | \( 1 - 579 T + 458114 T^{2} - 164950207 T^{3} + 76571147108 T^{4} - 20478126166699 T^{5} + 7388354639131788 T^{6} - 20478126166699 p^{3} T^{7} + 76571147108 p^{6} T^{8} - 164950207 p^{9} T^{9} + 458114 p^{12} T^{10} - 579 p^{15} T^{11} + p^{18} T^{12} \) | |
| 47 | \( 1 - 975 T + 920618 T^{2} - 510331091 T^{3} + 270532387624 T^{4} - 103646566472559 T^{5} + 38411384666106032 T^{6} - 103646566472559 p^{3} T^{7} + 270532387624 p^{6} T^{8} - 510331091 p^{9} T^{9} + 920618 p^{12} T^{10} - 975 p^{15} T^{11} + p^{18} T^{12} \) | |
| 53 | \( 1 - 515 T + 810798 T^{2} - 344250125 T^{3} + 288825389172 T^{4} - 96905829932359 T^{5} + 56620102961978056 T^{6} - 96905829932359 p^{3} T^{7} + 288825389172 p^{6} T^{8} - 344250125 p^{9} T^{9} + 810798 p^{12} T^{10} - 515 p^{15} T^{11} + p^{18} T^{12} \) | |
| 59 | \( 1 - 108 T + 1046870 T^{2} - 106050324 T^{3} + 488748295607 T^{4} - 42616006842040 T^{5} + 129658099730352500 T^{6} - 42616006842040 p^{3} T^{7} + 488748295607 p^{6} T^{8} - 106050324 p^{9} T^{9} + 1046870 p^{12} T^{10} - 108 p^{15} T^{11} + p^{18} T^{12} \) | |
| 61 | \( 1 - 1158 T + 1202234 T^{2} - 906701214 T^{3} + 600984155943 T^{4} - 347956687903124 T^{5} + 176426548565299372 T^{6} - 347956687903124 p^{3} T^{7} + 600984155943 p^{6} T^{8} - 906701214 p^{9} T^{9} + 1202234 p^{12} T^{10} - 1158 p^{15} T^{11} + p^{18} T^{12} \) | |
| 67 | \( 1 - 80 T + 915074 T^{2} - 113511984 T^{3} + 425550589879 T^{4} - 90804809236000 T^{5} + 2171969619668820 p T^{6} - 90804809236000 p^{3} T^{7} + 425550589879 p^{6} T^{8} - 113511984 p^{9} T^{9} + 915074 p^{12} T^{10} - 80 p^{15} T^{11} + p^{18} T^{12} \) | |
| 71 | \( 1 + 438 T + 393906 T^{2} + 161687442 T^{3} + 386934121583 T^{4} + 114013063753804 T^{5} + 100902934089145436 T^{6} + 114013063753804 p^{3} T^{7} + 386934121583 p^{6} T^{8} + 161687442 p^{9} T^{9} + 393906 p^{12} T^{10} + 438 p^{15} T^{11} + p^{18} T^{12} \) | |
| 73 | \( 1 - 262 T + 1080638 T^{2} - 754731406 T^{3} + 663831769535 T^{4} - 515007566749532 T^{5} + 328707154538754724 T^{6} - 515007566749532 p^{3} T^{7} + 663831769535 p^{6} T^{8} - 754731406 p^{9} T^{9} + 1080638 p^{12} T^{10} - 262 p^{15} T^{11} + p^{18} T^{12} \) | |
| 79 | \( 1 - 3 p T + 2439642 T^{2} - 484767321 T^{3} + 2683820153768 T^{4} - 432663133342445 T^{5} + 1698970446242928704 T^{6} - 432663133342445 p^{3} T^{7} + 2683820153768 p^{6} T^{8} - 484767321 p^{9} T^{9} + 2439642 p^{12} T^{10} - 3 p^{16} T^{11} + p^{18} T^{12} \) | |
| 83 | \( 1 - 1288 T + 2318470 T^{2} - 2159297064 T^{3} + 26893075837 p T^{4} - 1821786312002368 T^{5} + 1457826826827324244 T^{6} - 1821786312002368 p^{3} T^{7} + 26893075837 p^{7} T^{8} - 2159297064 p^{9} T^{9} + 2318470 p^{12} T^{10} - 1288 p^{15} T^{11} + p^{18} T^{12} \) | |
| 89 | \( 1 + 252 T + 2639798 T^{2} + 778547484 T^{3} + 3617328470639 T^{4} + 945395407751240 T^{5} + 3178085778118146644 T^{6} + 945395407751240 p^{3} T^{7} + 3617328470639 p^{6} T^{8} + 778547484 p^{9} T^{9} + 2639798 p^{12} T^{10} + 252 p^{15} T^{11} + p^{18} T^{12} \) | |
| 97 | \( 1 - 380 T + 2225910 T^{2} - 1028566556 T^{3} + 2470218411839 T^{4} - 1940510272261864 T^{5} + 2329611393484729876 T^{6} - 1940510272261864 p^{3} T^{7} + 2470218411839 p^{6} T^{8} - 1028566556 p^{9} T^{9} + 2225910 p^{12} T^{10} - 380 p^{15} T^{11} + p^{18} T^{12} \) | |
| 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
−5.58050745842937386548218561308, −5.07280430076268283372826076425, −5.03790947166743225867898369961, −4.92451934236838920263132825208, −4.90243826300076985781367645859, −4.78056046775851048037451688826, −4.30814374252757273297760507749, −4.30464552565133918483812648211, −3.83015557513799491795358693786, −3.79365787086787891333962573369, −3.69718535886224845980997985940, −3.45933320478391738336514866637, −3.29324293095534832684497780756, −2.90494558710677976101099955602, −2.63664200589246495169604096559, −2.47396620346706384497081457529, −2.44066899804843221657930758098, −2.22117605503401840899516939008, −2.09556751318977651585644643909, −1.44226751517513007511778127626, −1.28823915109855211328267494736, −1.27631052857903594711743970837, −0.70244542020119001740132804345, −0.46170085493285577366668345428, −0.45891674694207435171515451351, 0.45891674694207435171515451351, 0.46170085493285577366668345428, 0.70244542020119001740132804345, 1.27631052857903594711743970837, 1.28823915109855211328267494736, 1.44226751517513007511778127626, 2.09556751318977651585644643909, 2.22117605503401840899516939008, 2.44066899804843221657930758098, 2.47396620346706384497081457529, 2.63664200589246495169604096559, 2.90494558710677976101099955602, 3.29324293095534832684497780756, 3.45933320478391738336514866637, 3.69718535886224845980997985940, 3.79365787086787891333962573369, 3.83015557513799491795358693786, 4.30464552565133918483812648211, 4.30814374252757273297760507749, 4.78056046775851048037451688826, 4.90243826300076985781367645859, 4.92451934236838920263132825208, 5.03790947166743225867898369961, 5.07280430076268283372826076425, 5.58050745842937386548218561308
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.