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
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 29^{6}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{18} \cdot 29^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(6.57846\times 10^{6}\) |
| Root analytic conductor: | \(3.69978\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(6\) |
| Selberg data: | \((12,\ 2^{18} \cdot 29^{6} ,\ ( \ : [3/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{5}{2})\) | 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 | \( 1 - 380 T + 2225910 T^{2} - 1028566556 T^{3} + 2470218411839 T^{4} - 1940510272261864 T^{5} + 2329611393484729876 T^{6} - 1940510272261864 p^{3} T^{7} + 2470218411839 p^{6} T^{8} - 1028566556 p^{9} T^{9} + 2225910 p^{12} T^{10} - 380 p^{15} T^{11} + p^{18} T^{12} \) | |
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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
−6.78135280706039653171364947195, −6.67510143728699757940854467102, −6.11320033551999764730783014715, −6.07787216659288735223994267530, −6.02761196230189798507563543778, −6.02098460231851183120496830961, −6.00014507737750860422168568334, −5.28030756618209529649749073228, −5.25367509086055781930868840532, −5.20966092793844270827605557014, −4.92084309024214025258196428414, −4.66187714618072984787876784051, −4.27940224045732902416928477174, −3.99702968586669491013997372259, −3.87058736784354124432079338901, −3.75650357885373024571257901051, −3.55352137787283329290729638923, −3.41484096226827148790926224139, −3.23526107551971975573730793037, −2.49761884886699164232881243738, −2.31127594077476749613463110215, −2.29131263198630856565433557600, −1.98761575668701869713115027569, −1.78124472793754694070390273561, −1.56073988704642362253496501383, 0, 0, 0, 0, 0, 0, 1.56073988704642362253496501383, 1.78124472793754694070390273561, 1.98761575668701869713115027569, 2.29131263198630856565433557600, 2.31127594077476749613463110215, 2.49761884886699164232881243738, 3.23526107551971975573730793037, 3.41484096226827148790926224139, 3.55352137787283329290729638923, 3.75650357885373024571257901051, 3.87058736784354124432079338901, 3.99702968586669491013997372259, 4.27940224045732902416928477174, 4.66187714618072984787876784051, 4.92084309024214025258196428414, 5.20966092793844270827605557014, 5.25367509086055781930868840532, 5.28030756618209529649749073228, 6.00014507737750860422168568334, 6.02098460231851183120496830961, 6.02761196230189798507563543778, 6.07787216659288735223994267530, 6.11320033551999764730783014715, 6.67510143728699757940854467102, 6.78135280706039653171364947195
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.