Dirichlet series
| L(s) = 1 | − 4·2-s + 4-s − 7·5-s + 5·7-s + 15·8-s + 28·10-s + 13·11-s − 9·13-s − 20·14-s − 13·16-s − 7·17-s + 2·19-s − 7·20-s − 52·22-s − 23·23-s − 3·25-s + 36·26-s + 5·28-s − 16·29-s + 9·31-s − 28·32-s + 28·34-s − 35·35-s + 14·37-s − 8·38-s − 105·40-s − 19·41-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 1/2·4-s − 3.13·5-s + 1.88·7-s + 5.30·8-s + 8.85·10-s + 3.91·11-s − 2.49·13-s − 5.34·14-s − 3.25·16-s − 1.69·17-s + 0.458·19-s − 1.56·20-s − 11.0·22-s − 4.79·23-s − 3/5·25-s + 7.06·26-s + 0.944·28-s − 2.97·29-s + 1.61·31-s − 4.94·32-s + 4.80·34-s − 5.91·35-s + 2.30·37-s − 1.29·38-s − 16.6·40-s − 2.96·41-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: | \(13\) |
| Selberg data: | \((26,\ 3^{26} \cdot 11^{13} \cdot 61^{13} ,\ ( \ : [1/2]^{13} ),\ -1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 + p^{2} T + 15 T^{2} + 41 T^{3} + 51 p T^{4} + 111 p T^{5} + 457 T^{6} + 431 p T^{7} + 1569 T^{8} + 667 p^{2} T^{9} + 1097 p^{2} T^{10} + 3409 p T^{11} + 2571 p^{2} T^{12} + 14739 T^{13} + 2571 p^{3} T^{14} + 3409 p^{3} T^{15} + 1097 p^{5} T^{16} + 667 p^{6} T^{17} + 1569 p^{5} T^{18} + 431 p^{7} T^{19} + 457 p^{7} T^{20} + 111 p^{9} T^{21} + 51 p^{10} T^{22} + 41 p^{10} T^{23} + 15 p^{11} T^{24} + p^{14} T^{25} + p^{13} T^{26} \) |
| 5 | \( 1 + 7 T + 52 T^{2} + 2 p^{3} T^{3} + 1179 T^{4} + 4487 T^{5} + 16623 T^{6} + 53521 T^{7} + 167951 T^{8} + 94708 p T^{9} + 1304029 T^{10} + 3279296 T^{11} + 8063973 T^{12} + 18231446 T^{13} + 8063973 p T^{14} + 3279296 p^{2} T^{15} + 1304029 p^{3} T^{16} + 94708 p^{5} T^{17} + 167951 p^{5} T^{18} + 53521 p^{6} T^{19} + 16623 p^{7} T^{20} + 4487 p^{8} T^{21} + 1179 p^{9} T^{22} + 2 p^{13} T^{23} + 52 p^{11} T^{24} + 7 p^{12} T^{25} + p^{13} T^{26} \) | |
| 7 | \( 1 - 5 T + 54 T^{2} - 211 T^{3} + 192 p T^{4} - 4456 T^{5} + 21864 T^{6} - 64578 T^{7} + 269196 T^{8} - 723175 T^{9} + 2673431 T^{10} - 6578374 T^{11} + 22087536 T^{12} - 50044702 T^{13} + 22087536 p T^{14} - 6578374 p^{2} T^{15} + 2673431 p^{3} T^{16} - 723175 p^{4} T^{17} + 269196 p^{5} T^{18} - 64578 p^{6} T^{19} + 21864 p^{7} T^{20} - 4456 p^{8} T^{21} + 192 p^{10} T^{22} - 211 p^{10} T^{23} + 54 p^{11} T^{24} - 5 p^{12} T^{25} + p^{13} T^{26} \) | |
| 13 | \( 1 + 9 T + 126 T^{2} + 69 p T^{3} + 7314 T^{4} + 43127 T^{5} + 266094 T^{6} + 103158 p T^{7} + 6894558 T^{8} + 30470425 T^{9} + 136685974 T^{10} + 540624532 T^{11} + 2168911565 T^{12} + 7765534072 T^{13} + 2168911565 p T^{14} + 540624532 p^{2} T^{15} + 136685974 p^{3} T^{16} + 30470425 p^{4} T^{17} + 6894558 p^{5} T^{18} + 103158 p^{7} T^{19} + 266094 p^{7} T^{20} + 43127 p^{8} T^{21} + 7314 p^{9} T^{22} + 69 p^{11} T^{23} + 126 p^{11} T^{24} + 9 p^{12} T^{25} + p^{13} T^{26} \) | |
| 17 | \( 1 + 7 T + 172 T^{2} + 1134 T^{3} + 14517 T^{4} + 87622 T^{5} + 787227 T^{6} + 4278285 T^{7} + 30327181 T^{8} + 147154624 T^{9} + 871359928 T^{10} + 3756415546 T^{11} + 19150454978 T^{12} + 72946960856 T^{13} + 19150454978 p T^{14} + 3756415546 p^{2} T^{15} + 871359928 p^{3} T^{16} + 147154624 p^{4} T^{17} + 30327181 p^{5} T^{18} + 4278285 p^{6} T^{19} + 787227 p^{7} T^{20} + 87622 p^{8} T^{21} + 14517 p^{9} T^{22} + 1134 p^{10} T^{23} + 172 p^{11} T^{24} + 7 p^{12} T^{25} + p^{13} T^{26} \) | |
| 19 | \( 1 - 2 T + 158 T^{2} - 284 T^{3} + 12041 T^{4} - 19844 T^{5} + 594592 T^{6} - 900328 T^{7} + 21505328 T^{8} - 29700114 T^{9} + 608699913 T^{10} - 762713263 T^{11} + 13995831467 T^{12} - 15936313000 T^{13} + 13995831467 p T^{14} - 762713263 p^{2} T^{15} + 608699913 p^{3} T^{16} - 29700114 p^{4} T^{17} + 21505328 p^{5} T^{18} - 900328 p^{6} T^{19} + 594592 p^{7} T^{20} - 19844 p^{8} T^{21} + 12041 p^{9} T^{22} - 284 p^{10} T^{23} + 158 p^{11} T^{24} - 2 p^{12} T^{25} + p^{13} T^{26} \) | |
| 23 | \( 1 + p T + 405 T^{2} + 5212 T^{3} + 57841 T^{4} + 550153 T^{5} + 4713705 T^{6} + 36379173 T^{7} + 258356891 T^{8} + 1689249269 T^{9} + 10279724648 T^{10} + 58203767172 T^{11} + 308458660933 T^{12} + 1527720335132 T^{13} + 308458660933 p T^{14} + 58203767172 p^{2} T^{15} + 10279724648 p^{3} T^{16} + 1689249269 p^{4} T^{17} + 258356891 p^{5} T^{18} + 36379173 p^{6} T^{19} + 4713705 p^{7} T^{20} + 550153 p^{8} T^{21} + 57841 p^{9} T^{22} + 5212 p^{10} T^{23} + 405 p^{11} T^{24} + p^{13} T^{25} + p^{13} T^{26} \) | |
| 29 | \( 1 + 16 T + 339 T^{2} + 3906 T^{3} + 49572 T^{4} + 460161 T^{5} + 4444691 T^{6} + 35015453 T^{7} + 280272252 T^{8} + 1924075768 T^{9} + 13277547428 T^{10} + 80523088202 T^{11} + 488644075869 T^{12} + 2632299221416 T^{13} + 488644075869 p T^{14} + 80523088202 p^{2} T^{15} + 13277547428 p^{3} T^{16} + 1924075768 p^{4} T^{17} + 280272252 p^{5} T^{18} + 35015453 p^{6} T^{19} + 4444691 p^{7} T^{20} + 460161 p^{8} T^{21} + 49572 p^{9} T^{22} + 3906 p^{10} T^{23} + 339 p^{11} T^{24} + 16 p^{12} T^{25} + p^{13} T^{26} \) | |
| 31 | \( 1 - 9 T + 244 T^{2} - 1906 T^{3} + 27879 T^{4} - 201154 T^{5} + 2080402 T^{6} - 14398334 T^{7} + 116789053 T^{8} - 780130930 T^{9} + 5250384513 T^{10} - 33363036399 T^{11} + 194535543362 T^{12} - 1148596132950 T^{13} + 194535543362 p T^{14} - 33363036399 p^{2} T^{15} + 5250384513 p^{3} T^{16} - 780130930 p^{4} T^{17} + 116789053 p^{5} T^{18} - 14398334 p^{6} T^{19} + 2080402 p^{7} T^{20} - 201154 p^{8} T^{21} + 27879 p^{9} T^{22} - 1906 p^{10} T^{23} + 244 p^{11} T^{24} - 9 p^{12} T^{25} + p^{13} T^{26} \) | |
| 37 | \( 1 - 14 T + 415 T^{2} - 4761 T^{3} + 79829 T^{4} - 777369 T^{5} + 9566800 T^{6} - 80759411 T^{7} + 803041239 T^{8} - 5948863461 T^{9} + 50052301916 T^{10} - 327293775093 T^{11} + 2387786137488 T^{12} - 13787787297494 T^{13} + 2387786137488 p T^{14} - 327293775093 p^{2} T^{15} + 50052301916 p^{3} T^{16} - 5948863461 p^{4} T^{17} + 803041239 p^{5} T^{18} - 80759411 p^{6} T^{19} + 9566800 p^{7} T^{20} - 777369 p^{8} T^{21} + 79829 p^{9} T^{22} - 4761 p^{10} T^{23} + 415 p^{11} T^{24} - 14 p^{12} T^{25} + p^{13} T^{26} \) | |
| 41 | \( 1 + 19 T + 570 T^{2} + 8078 T^{3} + 140623 T^{4} + 1604457 T^{5} + 20664839 T^{6} + 197699421 T^{7} + 2057334849 T^{8} + 16890207028 T^{9} + 148144053509 T^{10} + 1056883015886 T^{11} + 7987872705573 T^{12} + 49754457667820 T^{13} + 7987872705573 p T^{14} + 1056883015886 p^{2} T^{15} + 148144053509 p^{3} T^{16} + 16890207028 p^{4} T^{17} + 2057334849 p^{5} T^{18} + 197699421 p^{6} T^{19} + 20664839 p^{7} T^{20} + 1604457 p^{8} T^{21} + 140623 p^{9} T^{22} + 8078 p^{10} T^{23} + 570 p^{11} T^{24} + 19 p^{12} T^{25} + p^{13} T^{26} \) | |
| 43 | \( 1 - 7 T + 411 T^{2} - 3130 T^{3} + 81725 T^{4} - 651656 T^{5} + 10451069 T^{6} - 84087081 T^{7} + 960506150 T^{8} - 7525014680 T^{9} + 66998871224 T^{10} - 493515264974 T^{11} + 3647340651788 T^{12} - 24361840908780 T^{13} + 3647340651788 p T^{14} - 493515264974 p^{2} T^{15} + 66998871224 p^{3} T^{16} - 7525014680 p^{4} T^{17} + 960506150 p^{5} T^{18} - 84087081 p^{6} T^{19} + 10451069 p^{7} T^{20} - 651656 p^{8} T^{21} + 81725 p^{9} T^{22} - 3130 p^{10} T^{23} + 411 p^{11} T^{24} - 7 p^{12} T^{25} + p^{13} T^{26} \) | |
| 47 | \( 1 + 26 T + 650 T^{2} + 11372 T^{3} + 180271 T^{4} + 2414607 T^{5} + 29737747 T^{6} + 326628724 T^{7} + 3338580611 T^{8} + 31177483834 T^{9} + 5803680626 p T^{10} + 2206112519221 T^{11} + 16781305763628 T^{12} + 2522261294908 p T^{13} + 16781305763628 p T^{14} + 2206112519221 p^{2} T^{15} + 5803680626 p^{4} T^{16} + 31177483834 p^{4} T^{17} + 3338580611 p^{5} T^{18} + 326628724 p^{6} T^{19} + 29737747 p^{7} T^{20} + 2414607 p^{8} T^{21} + 180271 p^{9} T^{22} + 11372 p^{10} T^{23} + 650 p^{11} T^{24} + 26 p^{12} T^{25} + p^{13} T^{26} \) | |
| 53 | \( 1 + 18 T + 397 T^{2} + 5159 T^{3} + 69779 T^{4} + 685382 T^{5} + 6697901 T^{6} + 48698795 T^{7} + 339648838 T^{8} + 1360288879 T^{9} + 2739046927 T^{10} - 66090461863 T^{11} - 752023040015 T^{12} - 7444225952930 T^{13} - 752023040015 p T^{14} - 66090461863 p^{2} T^{15} + 2739046927 p^{3} T^{16} + 1360288879 p^{4} T^{17} + 339648838 p^{5} T^{18} + 48698795 p^{6} T^{19} + 6697901 p^{7} T^{20} + 685382 p^{8} T^{21} + 69779 p^{9} T^{22} + 5159 p^{10} T^{23} + 397 p^{11} T^{24} + 18 p^{12} T^{25} + p^{13} T^{26} \) | |
| 59 | \( 1 + 31 T + 758 T^{2} + 11717 T^{3} + 161239 T^{4} + 1691848 T^{5} + 17364310 T^{6} + 146609274 T^{7} + 1343833743 T^{8} + 10554092416 T^{9} + 97314500021 T^{10} + 763530615380 T^{11} + 6882032222794 T^{12} + 50283233012916 T^{13} + 6882032222794 p T^{14} + 763530615380 p^{2} T^{15} + 97314500021 p^{3} T^{16} + 10554092416 p^{4} T^{17} + 1343833743 p^{5} T^{18} + 146609274 p^{6} T^{19} + 17364310 p^{7} T^{20} + 1691848 p^{8} T^{21} + 161239 p^{9} T^{22} + 11717 p^{10} T^{23} + 758 p^{11} T^{24} + 31 p^{12} T^{25} + p^{13} T^{26} \) | |
| 67 | \( 1 - 14 T + 462 T^{2} - 5387 T^{3} + 105068 T^{4} - 1066907 T^{5} + 15781773 T^{6} - 145190299 T^{7} + 1792731622 T^{8} - 15258806275 T^{9} + 165152548835 T^{10} - 1313267597677 T^{11} + 12847016504773 T^{12} - 95311824122652 T^{13} + 12847016504773 p T^{14} - 1313267597677 p^{2} T^{15} + 165152548835 p^{3} T^{16} - 15258806275 p^{4} T^{17} + 1792731622 p^{5} T^{18} - 145190299 p^{6} T^{19} + 15781773 p^{7} T^{20} - 1066907 p^{8} T^{21} + 105068 p^{9} T^{22} - 5387 p^{10} T^{23} + 462 p^{11} T^{24} - 14 p^{12} T^{25} + p^{13} T^{26} \) | |
| 71 | \( 1 + 37 T + 901 T^{2} + 14746 T^{3} + 206290 T^{4} + 2506621 T^{5} + 30453105 T^{6} + 345876887 T^{7} + 3803389784 T^{8} + 37856060224 T^{9} + 366954231344 T^{10} + 3367335479732 T^{11} + 30842666414903 T^{12} + 263798235920490 T^{13} + 30842666414903 p T^{14} + 3367335479732 p^{2} T^{15} + 366954231344 p^{3} T^{16} + 37856060224 p^{4} T^{17} + 3803389784 p^{5} T^{18} + 345876887 p^{6} T^{19} + 30453105 p^{7} T^{20} + 2506621 p^{8} T^{21} + 206290 p^{9} T^{22} + 14746 p^{10} T^{23} + 901 p^{11} T^{24} + 37 p^{12} T^{25} + p^{13} T^{26} \) | |
| 73 | \( 1 + 16 T + 630 T^{2} + 8528 T^{3} + 185642 T^{4} + 2185728 T^{5} + 34781245 T^{6} + 364567989 T^{7} + 4741032295 T^{8} + 45021431638 T^{9} + 506918130739 T^{10} + 4405175370490 T^{11} + 44355380310892 T^{12} + 353337388110734 T^{13} + 44355380310892 p T^{14} + 4405175370490 p^{2} T^{15} + 506918130739 p^{3} T^{16} + 45021431638 p^{4} T^{17} + 4741032295 p^{5} T^{18} + 364567989 p^{6} T^{19} + 34781245 p^{7} T^{20} + 2185728 p^{8} T^{21} + 185642 p^{9} T^{22} + 8528 p^{10} T^{23} + 630 p^{11} T^{24} + 16 p^{12} T^{25} + p^{13} T^{26} \) | |
| 79 | \( 1 + 17 T + 622 T^{2} + 8010 T^{3} + 182593 T^{4} + 1960260 T^{5} + 35114164 T^{6} + 328234756 T^{7} + 5053274170 T^{8} + 42318143615 T^{9} + 581709871210 T^{10} + 4424991480327 T^{11} + 55189321376644 T^{12} + 383304844703314 T^{13} + 55189321376644 p T^{14} + 4424991480327 p^{2} T^{15} + 581709871210 p^{3} T^{16} + 42318143615 p^{4} T^{17} + 5053274170 p^{5} T^{18} + 328234756 p^{6} T^{19} + 35114164 p^{7} T^{20} + 1960260 p^{8} T^{21} + 182593 p^{9} T^{22} + 8010 p^{10} T^{23} + 622 p^{11} T^{24} + 17 p^{12} T^{25} + p^{13} T^{26} \) | |
| 83 | \( 1 + 30 T + 974 T^{2} + 18458 T^{3} + 366294 T^{4} + 5398142 T^{5} + 83760451 T^{6} + 1050757230 T^{7} + 13881448384 T^{8} + 153604165536 T^{9} + 1782849294068 T^{10} + 17626677690403 T^{11} + 182574569560402 T^{12} + 1624662755159746 T^{13} + 182574569560402 p T^{14} + 17626677690403 p^{2} T^{15} + 1782849294068 p^{3} T^{16} + 153604165536 p^{4} T^{17} + 13881448384 p^{5} T^{18} + 1050757230 p^{6} T^{19} + 83760451 p^{7} T^{20} + 5398142 p^{8} T^{21} + 366294 p^{9} T^{22} + 18458 p^{10} T^{23} + 974 p^{11} T^{24} + 30 p^{12} T^{25} + p^{13} T^{26} \) | |
| 89 | \( 1 + 35 T + 1184 T^{2} + 25669 T^{3} + 516942 T^{4} + 8370310 T^{5} + 127074742 T^{6} + 1696233422 T^{7} + 21565020894 T^{8} + 253402493441 T^{9} + 2858507560085 T^{10} + 30474939017346 T^{11} + 310586541225446 T^{12} + 3003263361737920 T^{13} + 310586541225446 p T^{14} + 30474939017346 p^{2} T^{15} + 2858507560085 p^{3} T^{16} + 253402493441 p^{4} T^{17} + 21565020894 p^{5} T^{18} + 1696233422 p^{6} T^{19} + 127074742 p^{7} T^{20} + 8370310 p^{8} T^{21} + 516942 p^{9} T^{22} + 25669 p^{10} T^{23} + 1184 p^{11} T^{24} + 35 p^{12} T^{25} + p^{13} T^{26} \) | |
| 97 | \( 1 + T + 606 T^{2} - 479 T^{3} + 180587 T^{4} - 591622 T^{5} + 35070746 T^{6} - 224773222 T^{7} + 5007271033 T^{8} - 50444281573 T^{9} + 575603368072 T^{10} - 7793122388836 T^{11} + 58515039743159 T^{12} - 878038609738246 T^{13} + 58515039743159 p T^{14} - 7793122388836 p^{2} T^{15} + 575603368072 p^{3} T^{16} - 50444281573 p^{4} T^{17} + 5007271033 p^{5} T^{18} - 224773222 p^{6} T^{19} + 35070746 p^{7} T^{20} - 591622 p^{8} T^{21} + 180587 p^{9} T^{22} - 479 p^{10} T^{23} + 606 p^{11} T^{24} + p^{12} T^{25} + p^{13} T^{26} \) | |
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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
−2.68196362627306643338606051678, −2.67968331895060533833606675426, −2.44681392735372586375667739747, −2.44361680749133517566395781969, −2.41689922938729550646049531448, −2.24946107185812047457833056016, −2.07935430902680508118757543321, −2.07608490819155204162756584497, −1.91829880081318649649063335747, −1.89012131649781892706712638102, −1.86855638297612403874351645967, −1.82230932933189428325565419532, −1.79254488262065839654605738093, −1.78749699965979590543039916086, −1.52117201510003304347618333341, −1.51019013050397404491615700598, −1.43747825257117947748529287446, −1.33719031040155797518863550923, −1.19359520793810345402548041058, −1.17910978375436342181464650404, −1.15739366113225922690346739511, −1.14510831013924597962284683978, −1.05965139833224351303061115140, −0.938646557756210787220456385655, −0.832388922203992140079034022956, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.832388922203992140079034022956, 0.938646557756210787220456385655, 1.05965139833224351303061115140, 1.14510831013924597962284683978, 1.15739366113225922690346739511, 1.17910978375436342181464650404, 1.19359520793810345402548041058, 1.33719031040155797518863550923, 1.43747825257117947748529287446, 1.51019013050397404491615700598, 1.52117201510003304347618333341, 1.78749699965979590543039916086, 1.79254488262065839654605738093, 1.82230932933189428325565419532, 1.86855638297612403874351645967, 1.89012131649781892706712638102, 1.91829880081318649649063335747, 2.07608490819155204162756584497, 2.07935430902680508118757543321, 2.24946107185812047457833056016, 2.41689922938729550646049531448, 2.44361680749133517566395781969, 2.44681392735372586375667739747, 2.67968331895060533833606675426, 2.68196362627306643338606051678
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.