Dirichlet series
| L(s) = 1 | + 8·2-s + 36·4-s − 4·5-s + 4·7-s + 120·8-s − 32·10-s − 2·11-s + 10·13-s + 32·14-s + 330·16-s − 8·17-s + 16·19-s − 144·20-s − 16·22-s + 4·23-s + 80·26-s + 144·28-s − 8·29-s + 12·31-s + 792·32-s − 64·34-s − 16·35-s + 16·37-s + 128·38-s − 480·40-s − 16·41-s + 12·43-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 18·4-s − 1.78·5-s + 1.51·7-s + 42.4·8-s − 10.1·10-s − 0.603·11-s + 2.77·13-s + 8.55·14-s + 82.5·16-s − 1.94·17-s + 3.67·19-s − 32.1·20-s − 3.41·22-s + 0.834·23-s + 15.6·26-s + 27.2·28-s − 1.48·29-s + 2.15·31-s + 140.·32-s − 10.9·34-s − 2.70·35-s + 2.63·37-s + 20.7·38-s − 75.8·40-s − 2.49·41-s + 1.82·43-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 223^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 223^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{16} \cdot 223^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.11387\times 10^{12}\) |
| Root analytic conductor: | \(5.66144\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{16} \cdot 223^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(2529.919004\) |
| \(L(\frac12)\) | \(\approx\) | \(2529.919004\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T )^{8} \) |
| 3 | \( 1 \) | |
| 223 | \( ( 1 - T )^{8} \) | |
| good | 5 | \( 1 + 4 T + 16 T^{2} + 34 T^{3} + 22 p T^{4} + 238 T^{5} + 698 T^{6} + 1224 T^{7} + 3288 T^{8} + 1224 p T^{9} + 698 p^{2} T^{10} + 238 p^{3} T^{11} + 22 p^{5} T^{12} + 34 p^{5} T^{13} + 16 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - 4 T + 16 T^{2} - 20 T^{3} + 148 T^{4} - 468 T^{5} + 2080 T^{6} - 2820 T^{7} + 1354 p T^{8} - 2820 p T^{9} + 2080 p^{2} T^{10} - 468 p^{3} T^{11} + 148 p^{4} T^{12} - 20 p^{5} T^{13} + 16 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 2 T + 38 T^{2} + 32 T^{3} + 738 T^{4} + 304 T^{5} + 11720 T^{6} + 5706 T^{7} + 150992 T^{8} + 5706 p T^{9} + 11720 p^{2} T^{10} + 304 p^{3} T^{11} + 738 p^{4} T^{12} + 32 p^{5} T^{13} + 38 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 10 T + 94 T^{2} - 612 T^{3} + 3610 T^{4} - 18168 T^{5} + 83580 T^{6} - 345814 T^{7} + 1307872 T^{8} - 345814 p T^{9} + 83580 p^{2} T^{10} - 18168 p^{3} T^{11} + 3610 p^{4} T^{12} - 612 p^{5} T^{13} + 94 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 8 T + 72 T^{2} + 400 T^{3} + 2428 T^{4} + 11088 T^{5} + 54980 T^{6} + 222032 T^{7} + 1013314 T^{8} + 222032 p T^{9} + 54980 p^{2} T^{10} + 11088 p^{3} T^{11} + 2428 p^{4} T^{12} + 400 p^{5} T^{13} + 72 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 - 16 T + 8 p T^{2} - 1056 T^{3} + 6428 T^{4} - 34944 T^{5} + 178280 T^{6} - 847792 T^{7} + 3839846 T^{8} - 847792 p T^{9} + 178280 p^{2} T^{10} - 34944 p^{3} T^{11} + 6428 p^{4} T^{12} - 1056 p^{5} T^{13} + 8 p^{7} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 4 T + 48 T^{2} - 100 T^{3} + 1164 T^{4} - 692 T^{5} + 34832 T^{6} - 68628 T^{7} + 1062054 T^{8} - 68628 p T^{9} + 34832 p^{2} T^{10} - 692 p^{3} T^{11} + 1164 p^{4} T^{12} - 100 p^{5} T^{13} + 48 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 8 T + 184 T^{2} + 1256 T^{3} + 16060 T^{4} + 92664 T^{5} + 854216 T^{6} + 4129944 T^{7} + 30109862 T^{8} + 4129944 p T^{9} + 854216 p^{2} T^{10} + 92664 p^{3} T^{11} + 16060 p^{4} T^{12} + 1256 p^{5} T^{13} + 184 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 12 T + 180 T^{2} - 1292 T^{3} + 11692 T^{4} - 58972 T^{5} + 426416 T^{6} - 1697396 T^{7} + 12644594 T^{8} - 1697396 p T^{9} + 426416 p^{2} T^{10} - 58972 p^{3} T^{11} + 11692 p^{4} T^{12} - 1292 p^{5} T^{13} + 180 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 16 T + 272 T^{2} - 2720 T^{3} + 28236 T^{4} - 221248 T^{5} + 1805744 T^{6} - 11915088 T^{7} + 80272582 T^{8} - 11915088 p T^{9} + 1805744 p^{2} T^{10} - 221248 p^{3} T^{11} + 28236 p^{4} T^{12} - 2720 p^{5} T^{13} + 272 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 16 T + 8 p T^{2} + 3536 T^{3} + 42780 T^{4} + 356016 T^{5} + 3229304 T^{6} + 21927280 T^{7} + 160620166 T^{8} + 21927280 p T^{9} + 3229304 p^{2} T^{10} + 356016 p^{3} T^{11} + 42780 p^{4} T^{12} + 3536 p^{5} T^{13} + 8 p^{7} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 12 T + 160 T^{2} - 1628 T^{3} + 14796 T^{4} - 128092 T^{5} + 1004448 T^{6} - 7054892 T^{7} + 49383750 T^{8} - 7054892 p T^{9} + 1004448 p^{2} T^{10} - 128092 p^{3} T^{11} + 14796 p^{4} T^{12} - 1628 p^{5} T^{13} + 160 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 4 T + 136 T^{2} - 700 T^{3} + 13996 T^{4} - 64068 T^{5} + 950924 T^{6} - 4077972 T^{7} + 52261874 T^{8} - 4077972 p T^{9} + 950924 p^{2} T^{10} - 64068 p^{3} T^{11} + 13996 p^{4} T^{12} - 700 p^{5} T^{13} + 136 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 8 T + 328 T^{2} - 2232 T^{3} + 50076 T^{4} - 290568 T^{5} + 4697784 T^{6} - 23153656 T^{7} + 298519718 T^{8} - 23153656 p T^{9} + 4697784 p^{2} T^{10} - 290568 p^{3} T^{11} + 50076 p^{4} T^{12} - 2232 p^{5} T^{13} + 328 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 348 T^{2} - 102 T^{3} + 58026 T^{4} - 21534 T^{5} + 6032534 T^{6} - 2177220 T^{7} + 426474240 T^{8} - 2177220 p T^{9} + 6032534 p^{2} T^{10} - 21534 p^{3} T^{11} + 58026 p^{4} T^{12} - 102 p^{5} T^{13} + 348 p^{6} T^{14} + p^{8} T^{16} \) | |
| 61 | \( 1 - 26 T + 594 T^{2} - 8944 T^{3} + 123630 T^{4} - 1373152 T^{5} + 14358788 T^{6} - 2092974 p T^{7} + 1075004504 T^{8} - 2092974 p^{2} T^{9} + 14358788 p^{2} T^{10} - 1373152 p^{3} T^{11} + 123630 p^{4} T^{12} - 8944 p^{5} T^{13} + 594 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 10 T + 386 T^{2} + 40 p T^{3} + 66446 T^{4} + 5436 p T^{5} + 7427496 T^{6} + 34595050 T^{7} + 590663576 T^{8} + 34595050 p T^{9} + 7427496 p^{2} T^{10} + 5436 p^{4} T^{11} + 66446 p^{4} T^{12} + 40 p^{6} T^{13} + 386 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 8 T + 248 T^{2} + 1816 T^{3} + 33980 T^{4} + 226392 T^{5} + 3435336 T^{6} + 19612232 T^{7} + 269503430 T^{8} + 19612232 p T^{9} + 3435336 p^{2} T^{10} + 226392 p^{3} T^{11} + 33980 p^{4} T^{12} + 1816 p^{5} T^{13} + 248 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 8 T + 376 T^{2} - 2672 T^{3} + 70292 T^{4} - 439776 T^{5} + 8578872 T^{6} - 46802504 T^{7} + 739256326 T^{8} - 46802504 p T^{9} + 8578872 p^{2} T^{10} - 439776 p^{3} T^{11} + 70292 p^{4} T^{12} - 2672 p^{5} T^{13} + 376 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 16 T + 256 T^{2} - 3296 T^{3} + 50092 T^{4} - 519744 T^{5} + 5530816 T^{6} - 53429616 T^{7} + 543655078 T^{8} - 53429616 p T^{9} + 5530816 p^{2} T^{10} - 519744 p^{3} T^{11} + 50092 p^{4} T^{12} - 3296 p^{5} T^{13} + 256 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 44 T + 1184 T^{2} - 21900 T^{3} + 313388 T^{4} - 3561932 T^{5} + 34360992 T^{6} - 298331788 T^{7} + 2638545734 T^{8} - 298331788 p T^{9} + 34360992 p^{2} T^{10} - 3561932 p^{3} T^{11} + 313388 p^{4} T^{12} - 21900 p^{5} T^{13} + 1184 p^{6} T^{14} - 44 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 292 T^{2} - 784 T^{3} + 42156 T^{4} - 178408 T^{5} + 4613000 T^{6} - 19010800 T^{7} + 446258050 T^{8} - 19010800 p T^{9} + 4613000 p^{2} T^{10} - 178408 p^{3} T^{11} + 42156 p^{4} T^{12} - 784 p^{5} T^{13} + 292 p^{6} T^{14} + p^{8} T^{16} \) | |
| 97 | \( 1 - 12 T + 536 T^{2} - 3284 T^{3} + 94044 T^{4} + 59460 T^{5} + 5663016 T^{6} + 95080348 T^{7} + 130249670 T^{8} + 95080348 p T^{9} + 5663016 p^{2} T^{10} + 59460 p^{3} T^{11} + 94044 p^{4} T^{12} - 3284 p^{5} T^{13} + 536 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.61149739417453957263979721963, −3.31013133234727047140147955171, −3.27345483952252609031189137048, −3.22601611311323799164892757134, −3.18739385056813043617859494607, −2.97855727587697059664277518468, −2.97185588556206463885115398913, −2.92744210007617722298771774712, −2.84145075601348662891347374976, −2.40645107029951644121224027696, −2.36096224770803454338912785737, −2.34599701656992027923140804050, −2.18318174101065620002665803264, −2.01166449519229385263137595724, −1.92890887525876845993899743237, −1.87520132634679022791987996057, −1.85561723315999398616550888436, −1.46142672928273964458415285548, −1.34042352096795692805698721467, −1.05055052676853020913827300328, −0.976809463328849346527502799614, −0.964306229897430743459721361927, −0.886907967043702912198960064189, −0.54841338919487449272011150716, −0.40624662939065983175959824825, 0.40624662939065983175959824825, 0.54841338919487449272011150716, 0.886907967043702912198960064189, 0.964306229897430743459721361927, 0.976809463328849346527502799614, 1.05055052676853020913827300328, 1.34042352096795692805698721467, 1.46142672928273964458415285548, 1.85561723315999398616550888436, 1.87520132634679022791987996057, 1.92890887525876845993899743237, 2.01166449519229385263137595724, 2.18318174101065620002665803264, 2.34599701656992027923140804050, 2.36096224770803454338912785737, 2.40645107029951644121224027696, 2.84145075601348662891347374976, 2.92744210007617722298771774712, 2.97185588556206463885115398913, 2.97855727587697059664277518468, 3.18739385056813043617859494607, 3.22601611311323799164892757134, 3.27345483952252609031189137048, 3.31013133234727047140147955171, 3.61149739417453957263979721963
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.