Dirichlet series
| L(s) = 1 | − 2·2-s − 3·4-s − 3·5-s + 11·7-s + 7·8-s + 6·10-s − 13·11-s + 13·13-s − 22·14-s + 3·16-s − 17·17-s + 14·19-s + 9·20-s + 26·22-s − 7·23-s − 19·25-s − 26·26-s − 33·28-s + 6·29-s + 27·31-s − 4·32-s + 34·34-s − 33·35-s + 10·37-s − 28·38-s − 21·40-s − 3·41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s − 3/2·4-s − 1.34·5-s + 4.15·7-s + 2.47·8-s + 1.89·10-s − 3.91·11-s + 3.60·13-s − 5.87·14-s + 3/4·16-s − 4.12·17-s + 3.21·19-s + 2.01·20-s + 5.54·22-s − 1.45·23-s − 3.79·25-s − 5.09·26-s − 6.23·28-s + 1.11·29-s + 4.84·31-s − 0.707·32-s + 5.83·34-s − 5.57·35-s + 1.64·37-s − 4.54·38-s − 3.32·40-s − 0.468·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{26} \cdot 11^{13} \cdot 61^{13}\right)^{s/2} \, \Gamma_{\C}(s)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{26} \cdot 11^{13} \cdot 61^{13}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{13} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(26\) |
| Conductor: | \(3^{26} \cdot 11^{13} \cdot 61^{13}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.62338\times 10^{21}\) |
| Root analytic conductor: | \(6.94418\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((26,\ 3^{26} \cdot 11^{13} \cdot 61^{13} ,\ ( \ : [1/2]^{13} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(8.149266706\) |
| \(L(\frac12)\) | \(\approx\) | \(8.149266706\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 11 | \( ( 1 + T )^{13} \) | |
| 61 | \( ( 1 + T )^{13} \) | |
| good | 2 | \( 1 + p T + 7 T^{2} + 13 T^{3} + 15 p T^{4} + 3 p^{4} T^{5} + 87 T^{6} + 65 p T^{7} + 219 T^{8} + 5 p^{6} T^{9} + 33 p^{4} T^{10} + 381 p T^{11} + 151 p^{3} T^{12} + 1655 T^{13} + 151 p^{4} T^{14} + 381 p^{3} T^{15} + 33 p^{7} T^{16} + 5 p^{10} T^{17} + 219 p^{5} T^{18} + 65 p^{7} T^{19} + 87 p^{7} T^{20} + 3 p^{12} T^{21} + 15 p^{10} T^{22} + 13 p^{10} T^{23} + 7 p^{11} T^{24} + p^{13} T^{25} + p^{13} T^{26} \) |
| 5 | \( 1 + 3 T + 28 T^{2} + 76 T^{3} + 391 T^{4} + 207 p T^{5} + 3891 T^{6} + 10101 T^{7} + 31341 T^{8} + 77614 T^{9} + 214033 T^{10} + 494844 T^{11} + 1255819 T^{12} + 2680782 T^{13} + 1255819 p T^{14} + 494844 p^{2} T^{15} + 214033 p^{3} T^{16} + 77614 p^{4} T^{17} + 31341 p^{5} T^{18} + 10101 p^{6} T^{19} + 3891 p^{7} T^{20} + 207 p^{9} T^{21} + 391 p^{9} T^{22} + 76 p^{10} T^{23} + 28 p^{11} T^{24} + 3 p^{12} T^{25} + p^{13} T^{26} \) | |
| 7 | \( 1 - 11 T + 102 T^{2} - 671 T^{3} + 548 p T^{4} - 18778 T^{5} + 82582 T^{6} - 330300 T^{7} + 1216388 T^{8} - 4178141 T^{9} + 1919811 p T^{10} - 5822254 p T^{11} + 116913072 T^{12} - 317503782 T^{13} + 116913072 p T^{14} - 5822254 p^{3} T^{15} + 1919811 p^{4} T^{16} - 4178141 p^{4} T^{17} + 1216388 p^{5} T^{18} - 330300 p^{6} T^{19} + 82582 p^{7} T^{20} - 18778 p^{8} T^{21} + 548 p^{10} T^{22} - 671 p^{10} T^{23} + 102 p^{11} T^{24} - 11 p^{12} T^{25} + p^{13} T^{26} \) | |
| 13 | \( 1 - p T + 166 T^{2} - 1371 T^{3} + 834 p T^{4} - 68851 T^{5} + 422468 T^{6} - 2243530 T^{7} + 11585898 T^{8} - 53508605 T^{9} + 240926658 T^{10} - 985066146 T^{11} + 3931490743 T^{12} - 14343563288 T^{13} + 3931490743 p T^{14} - 985066146 p^{2} T^{15} + 240926658 p^{3} T^{16} - 53508605 p^{4} T^{17} + 11585898 p^{5} T^{18} - 2243530 p^{6} T^{19} + 422468 p^{7} T^{20} - 68851 p^{8} T^{21} + 834 p^{10} T^{22} - 1371 p^{10} T^{23} + 166 p^{11} T^{24} - p^{13} T^{25} + p^{13} T^{26} \) | |
| 17 | \( 1 + p T + 16 p T^{2} + 2926 T^{3} + 28825 T^{4} + 235090 T^{5} + 1768253 T^{6} + 11778123 T^{7} + 72933433 T^{8} + 411775206 T^{9} + 2173525244 T^{10} + 10605747960 T^{11} + 48539734912 T^{12} + 206539615464 T^{13} + 48539734912 p T^{14} + 10605747960 p^{2} T^{15} + 2173525244 p^{3} T^{16} + 411775206 p^{4} T^{17} + 72933433 p^{5} T^{18} + 11778123 p^{6} T^{19} + 1768253 p^{7} T^{20} + 235090 p^{8} T^{21} + 28825 p^{9} T^{22} + 2926 p^{10} T^{23} + 16 p^{12} T^{24} + p^{13} T^{25} + p^{13} T^{26} \) | |
| 19 | \( 1 - 14 T + 204 T^{2} - 1808 T^{3} + 15631 T^{4} - 105346 T^{5} + 685720 T^{6} - 3820068 T^{7} + 20509216 T^{8} - 99372182 T^{9} + 24639581 p T^{10} - 2085072809 T^{11} + 9228604441 T^{12} - 40078657088 T^{13} + 9228604441 p T^{14} - 2085072809 p^{2} T^{15} + 24639581 p^{4} T^{16} - 99372182 p^{4} T^{17} + 20509216 p^{5} T^{18} - 3820068 p^{6} T^{19} + 685720 p^{7} T^{20} - 105346 p^{8} T^{21} + 15631 p^{9} T^{22} - 1808 p^{10} T^{23} + 204 p^{11} T^{24} - 14 p^{12} T^{25} + p^{13} T^{26} \) | |
| 23 | \( 1 + 7 T + 173 T^{2} + 922 T^{3} + 13503 T^{4} + 2529 p T^{5} + 665521 T^{6} + 2367741 T^{7} + 23755639 T^{8} + 69941305 T^{9} + 671953888 T^{10} + 1681077876 T^{11} + 16541939243 T^{12} + 38191222924 T^{13} + 16541939243 p T^{14} + 1681077876 p^{2} T^{15} + 671953888 p^{3} T^{16} + 69941305 p^{4} T^{17} + 23755639 p^{5} T^{18} + 2367741 p^{6} T^{19} + 665521 p^{7} T^{20} + 2529 p^{9} T^{21} + 13503 p^{9} T^{22} + 922 p^{10} T^{23} + 173 p^{11} T^{24} + 7 p^{12} T^{25} + p^{13} T^{26} \) | |
| 29 | \( 1 - 6 T + 247 T^{2} - 1340 T^{3} + 30322 T^{4} - 5205 p T^{5} + 2453163 T^{6} - 11230145 T^{7} + 145432146 T^{8} - 610197940 T^{9} + 6649767014 T^{10} - 25405586634 T^{11} + 240773801099 T^{12} - 829277813728 T^{13} + 240773801099 p T^{14} - 25405586634 p^{2} T^{15} + 6649767014 p^{3} T^{16} - 610197940 p^{4} T^{17} + 145432146 p^{5} T^{18} - 11230145 p^{6} T^{19} + 2453163 p^{7} T^{20} - 5205 p^{9} T^{21} + 30322 p^{9} T^{22} - 1340 p^{10} T^{23} + 247 p^{11} T^{24} - 6 p^{12} T^{25} + p^{13} T^{26} \) | |
| 31 | \( 1 - 27 T + 600 T^{2} - 9126 T^{3} + 121595 T^{4} - 1335774 T^{5} + 13339118 T^{6} - 117336124 T^{7} + 960676419 T^{8} - 7155706592 T^{9} + 50280686263 T^{10} - 326325952633 T^{11} + 2008518015006 T^{12} - 11477446488818 T^{13} + 2008518015006 p T^{14} - 326325952633 p^{2} T^{15} + 50280686263 p^{3} T^{16} - 7155706592 p^{4} T^{17} + 960676419 p^{5} T^{18} - 117336124 p^{6} T^{19} + 13339118 p^{7} T^{20} - 1335774 p^{8} T^{21} + 121595 p^{9} T^{22} - 9126 p^{10} T^{23} + 600 p^{11} T^{24} - 27 p^{12} T^{25} + p^{13} T^{26} \) | |
| 37 | \( 1 - 10 T + 287 T^{2} - 2385 T^{3} + 39619 T^{4} - 285347 T^{5} + 3602946 T^{6} - 23124531 T^{7} + 245436067 T^{8} - 1428276167 T^{9} + 13329360568 T^{10} - 70866824157 T^{11} + 594175785360 T^{12} - 2885985951366 T^{13} + 594175785360 p T^{14} - 70866824157 p^{2} T^{15} + 13329360568 p^{3} T^{16} - 1428276167 p^{4} T^{17} + 245436067 p^{5} T^{18} - 23124531 p^{6} T^{19} + 3602946 p^{7} T^{20} - 285347 p^{8} T^{21} + 39619 p^{9} T^{22} - 2385 p^{10} T^{23} + 287 p^{11} T^{24} - 10 p^{12} T^{25} + p^{13} T^{26} \) | |
| 41 | \( 1 + 3 T + 326 T^{2} + 824 T^{3} + 52843 T^{4} + 112289 T^{5} + 5665641 T^{6} + 10220209 T^{7} + 450062925 T^{8} + 702745206 T^{9} + 28028911215 T^{10} + 38691561568 T^{11} + 1408022472661 T^{12} + 1746777455844 T^{13} + 1408022472661 p T^{14} + 38691561568 p^{2} T^{15} + 28028911215 p^{3} T^{16} + 702745206 p^{4} T^{17} + 450062925 p^{5} T^{18} + 10220209 p^{6} T^{19} + 5665641 p^{7} T^{20} + 112289 p^{8} T^{21} + 52843 p^{9} T^{22} + 824 p^{10} T^{23} + 326 p^{11} T^{24} + 3 p^{12} T^{25} + p^{13} T^{26} \) | |
| 43 | \( 1 - 29 T + 757 T^{2} - 13060 T^{3} + 205803 T^{4} - 2620114 T^{5} + 31060811 T^{6} - 318713121 T^{7} + 3089837128 T^{8} - 26804861612 T^{9} + 222110611870 T^{10} - 1677111262894 T^{11} + 12183450031082 T^{12} - 81345159312664 T^{13} + 12183450031082 p T^{14} - 1677111262894 p^{2} T^{15} + 222110611870 p^{3} T^{16} - 26804861612 p^{4} T^{17} + 3089837128 p^{5} T^{18} - 318713121 p^{6} T^{19} + 31060811 p^{7} T^{20} - 2620114 p^{8} T^{21} + 205803 p^{9} T^{22} - 13060 p^{10} T^{23} + 757 p^{11} T^{24} - 29 p^{12} T^{25} + p^{13} T^{26} \) | |
| 47 | \( 1 + 8 T + 382 T^{2} + 2706 T^{3} + 71179 T^{4} + 457541 T^{5} + 8750573 T^{6} + 51750872 T^{7} + 798880953 T^{8} + 4357243002 T^{9} + 57183188436 T^{10} + 286237174457 T^{11} + 70135107626 p T^{12} + 15005322289016 T^{13} + 70135107626 p^{2} T^{14} + 286237174457 p^{2} T^{15} + 57183188436 p^{3} T^{16} + 4357243002 p^{4} T^{17} + 798880953 p^{5} T^{18} + 51750872 p^{6} T^{19} + 8750573 p^{7} T^{20} + 457541 p^{8} T^{21} + 71179 p^{9} T^{22} + 2706 p^{10} T^{23} + 382 p^{11} T^{24} + 8 p^{12} T^{25} + p^{13} T^{26} \) | |
| 53 | \( 1 - 24 T + 475 T^{2} - 5611 T^{3} + 61605 T^{4} - 485724 T^{5} + 4243647 T^{6} - 28236809 T^{7} + 240196766 T^{8} - 1331226943 T^{9} + 10272475875 T^{10} - 37967197175 T^{11} + 334747029319 T^{12} - 1010333089226 T^{13} + 334747029319 p T^{14} - 37967197175 p^{2} T^{15} + 10272475875 p^{3} T^{16} - 1331226943 p^{4} T^{17} + 240196766 p^{5} T^{18} - 28236809 p^{6} T^{19} + 4243647 p^{7} T^{20} - 485724 p^{8} T^{21} + 61605 p^{9} T^{22} - 5611 p^{10} T^{23} + 475 p^{11} T^{24} - 24 p^{12} T^{25} + p^{13} T^{26} \) | |
| 59 | \( 1 + 13 T + 548 T^{2} + 5579 T^{3} + 133861 T^{4} + 1099286 T^{5} + 19937846 T^{6} + 134753940 T^{7} + 2096039823 T^{8} + 11909036154 T^{9} + 171318435787 T^{10} + 845692358222 T^{11} + 11668320878576 T^{12} + 52471781246340 T^{13} + 11668320878576 p T^{14} + 845692358222 p^{2} T^{15} + 171318435787 p^{3} T^{16} + 11909036154 p^{4} T^{17} + 2096039823 p^{5} T^{18} + 134753940 p^{6} T^{19} + 19937846 p^{7} T^{20} + 1099286 p^{8} T^{21} + 133861 p^{9} T^{22} + 5579 p^{10} T^{23} + 548 p^{11} T^{24} + 13 p^{12} T^{25} + p^{13} T^{26} \) | |
| 67 | \( 1 - 44 T + 1188 T^{2} - 24193 T^{3} + 411170 T^{4} - 6078465 T^{5} + 80297323 T^{6} - 965078585 T^{7} + 10701564022 T^{8} - 110643777683 T^{9} + 1075723589435 T^{10} - 9903635405709 T^{11} + 86805253524531 T^{12} - 726645176472648 T^{13} + 86805253524531 p T^{14} - 9903635405709 p^{2} T^{15} + 1075723589435 p^{3} T^{16} - 110643777683 p^{4} T^{17} + 10701564022 p^{5} T^{18} - 965078585 p^{6} T^{19} + 80297323 p^{7} T^{20} - 6078465 p^{8} T^{21} + 411170 p^{9} T^{22} - 24193 p^{10} T^{23} + 1188 p^{11} T^{24} - 44 p^{12} T^{25} + p^{13} T^{26} \) | |
| 71 | \( 1 + 3 T + 485 T^{2} + 1762 T^{3} + 117464 T^{4} + 493261 T^{5} + 19221703 T^{6} + 87958469 T^{7} + 2395386368 T^{8} + 11333779128 T^{9} + 240956043280 T^{10} + 1127862193268 T^{11} + 20201203996895 T^{12} + 89372246452338 T^{13} + 20201203996895 p T^{14} + 1127862193268 p^{2} T^{15} + 240956043280 p^{3} T^{16} + 11333779128 p^{4} T^{17} + 2395386368 p^{5} T^{18} + 87958469 p^{6} T^{19} + 19221703 p^{7} T^{20} + 493261 p^{8} T^{21} + 117464 p^{9} T^{22} + 1762 p^{10} T^{23} + 485 p^{11} T^{24} + 3 p^{12} T^{25} + p^{13} T^{26} \) | |
| 73 | \( 1 - 48 T + 1522 T^{2} - 34880 T^{3} + 653072 T^{4} - 10263838 T^{5} + 140801067 T^{6} - 1710105013 T^{7} + 18793932243 T^{8} - 189235230100 T^{9} + 1781965804351 T^{10} - 15946799449806 T^{11} + 138757519977572 T^{12} - 1188994646402494 T^{13} + 138757519977572 p T^{14} - 15946799449806 p^{2} T^{15} + 1781965804351 p^{3} T^{16} - 189235230100 p^{4} T^{17} + 18793932243 p^{5} T^{18} - 1710105013 p^{6} T^{19} + 140801067 p^{7} T^{20} - 10263838 p^{8} T^{21} + 653072 p^{9} T^{22} - 34880 p^{10} T^{23} + 1522 p^{11} T^{24} - 48 p^{12} T^{25} + p^{13} T^{26} \) | |
| 79 | \( 1 + 17 T + 542 T^{2} + 7530 T^{3} + 155435 T^{4} + 1879772 T^{5} + 30500780 T^{6} + 329653304 T^{7} + 4538991666 T^{8} + 44343627643 T^{9} + 536172795588 T^{10} + 4759796653037 T^{11} + 652267225316 p T^{12} + 415643920998238 T^{13} + 652267225316 p^{2} T^{14} + 4759796653037 p^{2} T^{15} + 536172795588 p^{3} T^{16} + 44343627643 p^{4} T^{17} + 4538991666 p^{5} T^{18} + 329653304 p^{6} T^{19} + 30500780 p^{7} T^{20} + 1879772 p^{8} T^{21} + 155435 p^{9} T^{22} + 7530 p^{10} T^{23} + 542 p^{11} T^{24} + 17 p^{12} T^{25} + p^{13} T^{26} \) | |
| 83 | \( 1 + 50 T + 1798 T^{2} + 46774 T^{3} + 1033384 T^{4} + 19372996 T^{5} + 325384261 T^{6} + 4885010702 T^{7} + 67314181298 T^{8} + 848528007260 T^{9} + 9938314273754 T^{10} + 107686123909759 T^{11} + 1090671424852278 T^{12} + 10268166394136662 T^{13} + 1090671424852278 p T^{14} + 107686123909759 p^{2} T^{15} + 9938314273754 p^{3} T^{16} + 848528007260 p^{4} T^{17} + 67314181298 p^{5} T^{18} + 4885010702 p^{6} T^{19} + 325384261 p^{7} T^{20} + 19372996 p^{8} T^{21} + 1033384 p^{9} T^{22} + 46774 p^{10} T^{23} + 1798 p^{11} T^{24} + 50 p^{12} T^{25} + p^{13} T^{26} \) | |
| 89 | \( 1 - 15 T + 758 T^{2} - 8993 T^{3} + 264074 T^{4} - 2598648 T^{5} + 58536502 T^{6} - 495288546 T^{7} + 9541671058 T^{8} - 71450887279 T^{9} + 1235582283815 T^{10} - 8354202545906 T^{11} + 131850752462226 T^{12} - 812884574460588 T^{13} + 131850752462226 p T^{14} - 8354202545906 p^{2} T^{15} + 1235582283815 p^{3} T^{16} - 71450887279 p^{4} T^{17} + 9541671058 p^{5} T^{18} - 495288546 p^{6} T^{19} + 58536502 p^{7} T^{20} - 2598648 p^{8} T^{21} + 264074 p^{9} T^{22} - 8993 p^{10} T^{23} + 758 p^{11} T^{24} - 15 p^{12} T^{25} + p^{13} T^{26} \) | |
| 97 | \( 1 - 27 T + 846 T^{2} - 16071 T^{3} + 316643 T^{4} - 4868078 T^{5} + 75911390 T^{6} - 1006444954 T^{7} + 13447787513 T^{8} - 158930900501 T^{9} + 1889630845088 T^{10} - 20281662413284 T^{11} + 218857804683067 T^{12} - 2150177092562598 T^{13} + 218857804683067 p T^{14} - 20281662413284 p^{2} T^{15} + 1889630845088 p^{3} T^{16} - 158930900501 p^{4} T^{17} + 13447787513 p^{5} T^{18} - 1006444954 p^{6} T^{19} + 75911390 p^{7} T^{20} - 4868078 p^{8} T^{21} + 316643 p^{9} T^{22} - 16071 p^{10} T^{23} + 846 p^{11} T^{24} - 27 p^{12} T^{25} + p^{13} T^{26} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{26} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.18982533363954486561705855382, −2.16149840244085666868950148157, −2.11793738331804301058112119051, −1.92539575037766351137464208310, −1.92293049102032235174648026791, −1.86587476640607329099681152317, −1.80919977223428335133748038831, −1.54993844702240346956756008628, −1.52743929085332522593763669464, −1.52022422365320208498467372728, −1.51667631295249208967555247252, −1.42015122165833679228723903519, −1.37425451499831577401015566464, −1.02388964045873747986552837651, −0.980646825568715714633671622017, −0.961207751784795193313491134735, −0.78627884101900068742061038523, −0.77113869456304046969718285908, −0.66351049452631776571306192733, −0.54295278966997822666812862566, −0.48463816212071062368203895018, −0.48212482915960218693547648457, −0.43224267904928466605158701461, −0.33480758345315629334503361709, −0.13395458072333980008735686771, 0.13395458072333980008735686771, 0.33480758345315629334503361709, 0.43224267904928466605158701461, 0.48212482915960218693547648457, 0.48463816212071062368203895018, 0.54295278966997822666812862566, 0.66351049452631776571306192733, 0.77113869456304046969718285908, 0.78627884101900068742061038523, 0.961207751784795193313491134735, 0.980646825568715714633671622017, 1.02388964045873747986552837651, 1.37425451499831577401015566464, 1.42015122165833679228723903519, 1.51667631295249208967555247252, 1.52022422365320208498467372728, 1.52743929085332522593763669464, 1.54993844702240346956756008628, 1.80919977223428335133748038831, 1.86587476640607329099681152317, 1.92293049102032235174648026791, 1.92539575037766351137464208310, 2.11793738331804301058112119051, 2.16149840244085666868950148157, 2.18982533363954486561705855382
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.