Dirichlet series
| L(s) = 1 | − 4·2-s + 4-s − 7·5-s + 7·7-s + 17·8-s + 28·10-s − 13·11-s + 9·13-s − 28·14-s − 25·16-s − 19·17-s + 14·19-s − 7·20-s + 52·22-s − 5·23-s − 7·25-s − 36·26-s + 7·28-s − 10·29-s − 31-s − 8·32-s + 76·34-s − 49·35-s − 8·37-s − 56·38-s − 119·40-s − 21·41-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 1/2·4-s − 3.13·5-s + 2.64·7-s + 6.01·8-s + 8.85·10-s − 3.91·11-s + 2.49·13-s − 7.48·14-s − 6.25·16-s − 4.60·17-s + 3.21·19-s − 1.56·20-s + 11.0·22-s − 1.04·23-s − 7/5·25-s − 7.06·26-s + 1.32·28-s − 1.85·29-s − 0.179·31-s − 1.41·32-s + 13.0·34-s − 8.28·35-s − 1.31·37-s − 9.08·38-s − 18.8·40-s − 3.27·41-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 | \( 1 + p^{2} T + 15 T^{2} + 39 T^{3} + 49 p T^{4} + 103 p T^{5} + 423 T^{6} + 97 p^{3} T^{7} + 1407 T^{8} + 147 p^{4} T^{9} + 243 p^{4} T^{10} + 375 p^{4} T^{11} + 4569 p T^{12} + 13027 T^{13} + 4569 p^{2} T^{14} + 375 p^{6} T^{15} + 243 p^{7} T^{16} + 147 p^{8} T^{17} + 1407 p^{5} T^{18} + 97 p^{9} T^{19} + 423 p^{7} T^{20} + 103 p^{9} T^{21} + 49 p^{10} T^{22} + 39 p^{10} T^{23} + 15 p^{11} T^{24} + p^{14} T^{25} + p^{13} T^{26} \) |
| 5 | \( 1 + 7 T + 56 T^{2} + 258 T^{3} + 251 p T^{4} + 4503 T^{5} + 16859 T^{6} + 10177 p T^{7} + 160423 T^{8} + 425008 T^{9} + 1180717 T^{10} + 2815444 T^{11} + 1414113 p T^{12} + 15390246 T^{13} + 1414113 p^{2} T^{14} + 2815444 p^{2} T^{15} + 1180717 p^{3} T^{16} + 425008 p^{4} T^{17} + 160423 p^{5} T^{18} + 10177 p^{7} T^{19} + 16859 p^{7} T^{20} + 4503 p^{8} T^{21} + 251 p^{10} T^{22} + 258 p^{10} T^{23} + 56 p^{11} T^{24} + 7 p^{12} T^{25} + p^{13} T^{26} \) | |
| 7 | \( 1 - p T + 10 p T^{2} - 369 T^{3} + 2288 T^{4} - 9950 T^{5} + 6806 p T^{6} - 25436 p T^{7} + 711086 T^{8} - 2332387 T^{9} + 8053671 T^{10} - 23432188 T^{11} + 71259100 T^{12} - 184614114 T^{13} + 71259100 p T^{14} - 23432188 p^{2} T^{15} + 8053671 p^{3} T^{16} - 2332387 p^{4} T^{17} + 711086 p^{5} T^{18} - 25436 p^{7} T^{19} + 6806 p^{8} T^{20} - 9950 p^{8} T^{21} + 2288 p^{9} T^{22} - 369 p^{10} T^{23} + 10 p^{12} T^{24} - p^{13} T^{25} + p^{13} T^{26} \) | |
| 13 | \( 1 - 9 T + 142 T^{2} - 963 T^{3} + 8890 T^{4} - 49469 T^{5} + 343406 T^{6} - 126238 p T^{7} + 9394370 T^{8} - 39521641 T^{9} + 14996206 p T^{10} - 730889714 T^{11} + 3178238861 T^{12} - 10658263028 T^{13} + 3178238861 p T^{14} - 730889714 p^{2} T^{15} + 14996206 p^{4} T^{16} - 39521641 p^{4} T^{17} + 9394370 p^{5} T^{18} - 126238 p^{7} T^{19} + 343406 p^{7} T^{20} - 49469 p^{8} T^{21} + 8890 p^{9} T^{22} - 963 p^{10} T^{23} + 142 p^{11} T^{24} - 9 p^{12} T^{25} + p^{13} T^{26} \) | |
| 17 | \( 1 + 19 T + 286 T^{2} + 3030 T^{3} + 27921 T^{4} + 216056 T^{5} + 1519141 T^{6} + 9542471 T^{7} + 55860051 T^{8} + 300909000 T^{9} + 1531481120 T^{10} + 7271209038 T^{11} + 32798268168 T^{12} + 138656460496 T^{13} + 32798268168 p T^{14} + 7271209038 p^{2} T^{15} + 1531481120 p^{3} T^{16} + 300909000 p^{4} T^{17} + 55860051 p^{5} T^{18} + 9542471 p^{6} T^{19} + 1519141 p^{7} T^{20} + 216056 p^{8} T^{21} + 27921 p^{9} T^{22} + 3030 p^{10} T^{23} + 286 p^{11} T^{24} + 19 p^{12} T^{25} + p^{13} T^{26} \) | |
| 19 | \( 1 - 14 T + 248 T^{2} - 2424 T^{3} + 25701 T^{4} - 198294 T^{5} + 1592044 T^{6} - 10285272 T^{7} + 68058462 T^{8} - 379790666 T^{9} + 2155393863 T^{10} - 10559591841 T^{11} + 52449577423 T^{12} - 227152092088 T^{13} + 52449577423 p T^{14} - 10559591841 p^{2} T^{15} + 2155393863 p^{3} T^{16} - 379790666 p^{4} T^{17} + 68058462 p^{5} T^{18} - 10285272 p^{6} T^{19} + 1592044 p^{7} T^{20} - 198294 p^{8} T^{21} + 25701 p^{9} T^{22} - 2424 p^{10} T^{23} + 248 p^{11} T^{24} - 14 p^{12} T^{25} + p^{13} T^{26} \) | |
| 23 | \( 1 + 5 T + 215 T^{2} + 1100 T^{3} + 22269 T^{4} + 114329 T^{5} + 1478099 T^{6} + 7463909 T^{7} + 70504439 T^{8} + 342737121 T^{9} + 2566070728 T^{10} + 11715614720 T^{11} + 73674321281 T^{12} + 306620883056 T^{13} + 73674321281 p T^{14} + 11715614720 p^{2} T^{15} + 2566070728 p^{3} T^{16} + 342737121 p^{4} T^{17} + 70504439 p^{5} T^{18} + 7463909 p^{6} T^{19} + 1478099 p^{7} T^{20} + 114329 p^{8} T^{21} + 22269 p^{9} T^{22} + 1100 p^{10} T^{23} + 215 p^{11} T^{24} + 5 p^{12} T^{25} + p^{13} T^{26} \) | |
| 29 | \( 1 + 10 T + 271 T^{2} + 80 p T^{3} + 34320 T^{4} + 250863 T^{5} + 2687247 T^{6} + 16896957 T^{7} + 146990240 T^{8} + 27860298 p T^{9} + 6107409596 T^{10} + 30083037390 T^{11} + 206972882469 T^{12} + 938460561856 T^{13} + 206972882469 p T^{14} + 30083037390 p^{2} T^{15} + 6107409596 p^{3} T^{16} + 27860298 p^{5} T^{17} + 146990240 p^{5} T^{18} + 16896957 p^{6} T^{19} + 2687247 p^{7} T^{20} + 250863 p^{8} T^{21} + 34320 p^{9} T^{22} + 80 p^{11} T^{23} + 271 p^{11} T^{24} + 10 p^{12} T^{25} + p^{13} T^{26} \) | |
| 31 | \( 1 + T + 288 T^{2} + 128 T^{3} + 40625 T^{4} + 2854 T^{5} + 3714170 T^{6} - 635768 T^{7} + 245031489 T^{8} - 75359216 T^{9} + 12295404095 T^{10} - 4425200321 T^{11} + 482112034156 T^{12} - 167128670114 T^{13} + 482112034156 p T^{14} - 4425200321 p^{2} T^{15} + 12295404095 p^{3} T^{16} - 75359216 p^{4} T^{17} + 245031489 p^{5} T^{18} - 635768 p^{6} T^{19} + 3714170 p^{7} T^{20} + 2854 p^{8} T^{21} + 40625 p^{9} T^{22} + 128 p^{10} T^{23} + 288 p^{11} T^{24} + p^{12} T^{25} + p^{13} T^{26} \) | |
| 37 | \( 1 + 8 T + 275 T^{2} + 1701 T^{3} + 34615 T^{4} + 165433 T^{5} + 2730912 T^{6} + 9899925 T^{7} + 157492443 T^{8} + 426504573 T^{9} + 7393508744 T^{10} + 15442255673 T^{11} + 303399072842 T^{12} + 552624316190 T^{13} + 303399072842 p T^{14} + 15442255673 p^{2} T^{15} + 7393508744 p^{3} T^{16} + 426504573 p^{4} T^{17} + 157492443 p^{5} T^{18} + 9899925 p^{6} T^{19} + 2730912 p^{7} T^{20} + 165433 p^{8} T^{21} + 34615 p^{9} T^{22} + 1701 p^{10} T^{23} + 275 p^{11} T^{24} + 8 p^{12} T^{25} + p^{13} T^{26} \) | |
| 41 | \( 1 + 21 T + 540 T^{2} + 7636 T^{3} + 116825 T^{4} + 1280643 T^{5} + 14810923 T^{6} + 135199551 T^{7} + 1296033361 T^{8} + 10264106178 T^{9} + 85277910821 T^{10} + 598271745082 T^{11} + 4396906526603 T^{12} + 27518809751956 T^{13} + 4396906526603 p T^{14} + 598271745082 p^{2} T^{15} + 85277910821 p^{3} T^{16} + 10264106178 p^{4} T^{17} + 1296033361 p^{5} T^{18} + 135199551 p^{6} T^{19} + 14810923 p^{7} T^{20} + 1280643 p^{8} T^{21} + 116825 p^{9} T^{22} + 7636 p^{10} T^{23} + 540 p^{11} T^{24} + 21 p^{12} T^{25} + p^{13} T^{26} \) | |
| 43 | \( 1 - 11 T + 343 T^{2} - 2734 T^{3} + 54061 T^{4} - 355546 T^{5} + 5728635 T^{6} - 33015751 T^{7} + 460695872 T^{8} - 2369020056 T^{9} + 29408375578 T^{10} - 136593716802 T^{11} + 1532816968042 T^{12} - 6469732775516 T^{13} + 1532816968042 p T^{14} - 136593716802 p^{2} T^{15} + 29408375578 p^{3} T^{16} - 2369020056 p^{4} T^{17} + 460695872 p^{5} T^{18} - 33015751 p^{6} T^{19} + 5728635 p^{7} T^{20} - 355546 p^{8} T^{21} + 54061 p^{9} T^{22} - 2734 p^{10} T^{23} + 343 p^{11} T^{24} - 11 p^{12} T^{25} + p^{13} T^{26} \) | |
| 47 | \( 1 + 22 T + 642 T^{2} + 10096 T^{3} + 174563 T^{4} + 2175831 T^{5} + 28224583 T^{6} + 293657148 T^{7} + 3111592527 T^{8} + 27806395138 T^{9} + 250636904182 T^{10} + 1953859409661 T^{11} + 15297994055552 T^{12} + 104712788542460 T^{13} + 15297994055552 p T^{14} + 1953859409661 p^{2} T^{15} + 250636904182 p^{3} T^{16} + 27806395138 p^{4} T^{17} + 3111592527 p^{5} T^{18} + 293657148 p^{6} T^{19} + 28224583 p^{7} T^{20} + 2175831 p^{8} T^{21} + 174563 p^{9} T^{22} + 10096 p^{10} T^{23} + 642 p^{11} T^{24} + 22 p^{12} T^{25} + p^{13} T^{26} \) | |
| 53 | \( 1 + 16 T + 703 T^{2} + 9307 T^{3} + 224125 T^{4} + 2520330 T^{5} + 43323009 T^{6} + 420329087 T^{7} + 5696644844 T^{8} + 48096154711 T^{9} + 539874071107 T^{10} + 3978515326043 T^{11} + 38008878473513 T^{12} + 243974894241370 T^{13} + 38008878473513 p T^{14} + 3978515326043 p^{2} T^{15} + 539874071107 p^{3} T^{16} + 48096154711 p^{4} T^{17} + 5696644844 p^{5} T^{18} + 420329087 p^{6} T^{19} + 43323009 p^{7} T^{20} + 2520330 p^{8} T^{21} + 224125 p^{9} T^{22} + 9307 p^{10} T^{23} + 703 p^{11} T^{24} + 16 p^{12} T^{25} + p^{13} T^{26} \) | |
| 59 | \( 1 + 19 T + 588 T^{2} + 8387 T^{3} + 150323 T^{4} + 1725136 T^{5} + 23246896 T^{6} + 226730256 T^{7} + 2557136489 T^{8} + 22130373950 T^{9} + 220805866149 T^{10} + 1736747698856 T^{11} + 15678767927188 T^{12} + 112576920082032 T^{13} + 15678767927188 p T^{14} + 1736747698856 p^{2} T^{15} + 220805866149 p^{3} T^{16} + 22130373950 p^{4} T^{17} + 2557136489 p^{5} T^{18} + 226730256 p^{6} T^{19} + 23246896 p^{7} T^{20} + 1725136 p^{8} T^{21} + 150323 p^{9} T^{22} + 8387 p^{10} T^{23} + 588 p^{11} T^{24} + 19 p^{12} T^{25} + p^{13} T^{26} \) | |
| 67 | \( 1 - 12 T + 476 T^{2} - 5497 T^{3} + 116752 T^{4} - 1262381 T^{5} + 19411279 T^{6} - 192068545 T^{7} + 2419350272 T^{8} - 21779688147 T^{9} + 238485194313 T^{10} - 1959319395389 T^{11} + 19212910079827 T^{12} - 144341874973268 T^{13} + 19212910079827 p T^{14} - 1959319395389 p^{2} T^{15} + 238485194313 p^{3} T^{16} - 21779688147 p^{4} T^{17} + 2419350272 p^{5} T^{18} - 192068545 p^{6} T^{19} + 19411279 p^{7} T^{20} - 1262381 p^{8} T^{21} + 116752 p^{9} T^{22} - 5497 p^{10} T^{23} + 476 p^{11} T^{24} - 12 p^{12} T^{25} + p^{13} T^{26} \) | |
| 71 | \( 1 + 5 T + 667 T^{2} + 2790 T^{3} + 210426 T^{4} + 721315 T^{5} + 41963243 T^{6} + 115239987 T^{7} + 5974373412 T^{8} + 12982883096 T^{9} + 650776182024 T^{10} + 1141711263752 T^{11} + 56611739724711 T^{12} + 85847839551638 T^{13} + 56611739724711 p T^{14} + 1141711263752 p^{2} T^{15} + 650776182024 p^{3} T^{16} + 12982883096 p^{4} T^{17} + 5974373412 p^{5} T^{18} + 115239987 p^{6} T^{19} + 41963243 p^{7} T^{20} + 721315 p^{8} T^{21} + 210426 p^{9} T^{22} + 2790 p^{10} T^{23} + 667 p^{11} T^{24} + 5 p^{12} T^{25} + p^{13} T^{26} \) | |
| 73 | \( 1 - 18 T + 722 T^{2} - 11164 T^{3} + 249956 T^{4} - 3378036 T^{5} + 55268985 T^{6} - 658789573 T^{7} + 8728233621 T^{8} - 92269947656 T^{9} + 1040846671509 T^{10} - 9788272584286 T^{11} + 96495170021146 T^{12} - 807214488838654 T^{13} + 96495170021146 p T^{14} - 9788272584286 p^{2} T^{15} + 1040846671509 p^{3} T^{16} - 92269947656 p^{4} T^{17} + 8728233621 p^{5} T^{18} - 658789573 p^{6} T^{19} + 55268985 p^{7} T^{20} - 3378036 p^{8} T^{21} + 249956 p^{9} T^{22} - 11164 p^{10} T^{23} + 722 p^{11} T^{24} - 18 p^{12} T^{25} + p^{13} T^{26} \) | |
| 79 | \( 1 + T + 592 T^{2} + 1682 T^{3} + 175369 T^{4} + 744808 T^{5} + 34871614 T^{6} + 181072780 T^{7} + 5201860224 T^{8} + 29584351201 T^{9} + 613761565080 T^{10} + 3530100791245 T^{11} + 58889113102736 T^{12} + 318873705758666 T^{13} + 58889113102736 p T^{14} + 3530100791245 p^{2} T^{15} + 613761565080 p^{3} T^{16} + 29584351201 p^{4} T^{17} + 5201860224 p^{5} T^{18} + 181072780 p^{6} T^{19} + 34871614 p^{7} T^{20} + 744808 p^{8} T^{21} + 175369 p^{9} T^{22} + 1682 p^{10} T^{23} + 592 p^{11} T^{24} + p^{12} T^{25} + p^{13} T^{26} \) | |
| 83 | \( 1 + 48 T + 1946 T^{2} + 54074 T^{3} + 1322286 T^{4} + 26651308 T^{5} + 485319219 T^{6} + 7734534804 T^{7} + 113100793526 T^{8} + 1486614970798 T^{9} + 18090592348036 T^{10} + 200478795669543 T^{11} + 2066582253204772 T^{12} + 19506872219703810 T^{13} + 2066582253204772 p T^{14} + 200478795669543 p^{2} T^{15} + 18090592348036 p^{3} T^{16} + 1486614970798 p^{4} T^{17} + 113100793526 p^{5} T^{18} + 7734534804 p^{6} T^{19} + 485319219 p^{7} T^{20} + 26651308 p^{8} T^{21} + 1322286 p^{9} T^{22} + 54074 p^{10} T^{23} + 1946 p^{11} T^{24} + 48 p^{12} T^{25} + p^{13} T^{26} \) | |
| 89 | \( 1 + 15 T + 786 T^{2} + 11565 T^{3} + 310984 T^{4} + 4296636 T^{5} + 81310600 T^{6} + 1023800184 T^{7} + 15480612414 T^{8} + 175057827139 T^{9} + 2244424047975 T^{10} + 22673108801304 T^{11} + 253614549212220 T^{12} + 2281130993837004 T^{13} + 253614549212220 p T^{14} + 22673108801304 p^{2} T^{15} + 2244424047975 p^{3} T^{16} + 175057827139 p^{4} T^{17} + 15480612414 p^{5} T^{18} + 1023800184 p^{6} T^{19} + 81310600 p^{7} T^{20} + 4296636 p^{8} T^{21} + 310984 p^{9} T^{22} + 11565 p^{10} T^{23} + 786 p^{11} T^{24} + 15 p^{12} T^{25} + p^{13} T^{26} \) | |
| 97 | \( 1 + 17 T + 938 T^{2} + 14141 T^{3} + 424719 T^{4} + 5744714 T^{5} + 123323034 T^{6} + 1505382982 T^{7} + 25668744301 T^{8} + 283066250203 T^{9} + 4047350409404 T^{10} + 40194221467436 T^{11} + 497446404474603 T^{12} + 4416178399171106 T^{13} + 497446404474603 p T^{14} + 40194221467436 p^{2} T^{15} + 4047350409404 p^{3} T^{16} + 283066250203 p^{4} T^{17} + 25668744301 p^{5} T^{18} + 1505382982 p^{6} T^{19} + 123323034 p^{7} T^{20} + 5744714 p^{8} T^{21} + 424719 p^{9} T^{22} + 14141 p^{10} T^{23} + 938 p^{11} T^{24} + 17 p^{12} T^{25} + p^{13} T^{26} \) | |
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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
−2.65623199979130886835599235353, −2.58090701112526895706712583342, −2.50387630881054782008760339597, −2.47297237950993398159363402502, −2.32246138668513310679957205749, −2.24537652181271127555958974599, −2.23749045348730158485048570554, −2.08391901213598401675018090560, −2.04804898273480102548309523599, −1.99997276896997845658011015985, −1.90626658867245593585975265043, −1.81329266254869614951347237511, −1.79019018887551384872374682982, −1.75749280291071720437399199608, −1.53927324392953859261830940732, −1.48755758827704879203405734711, −1.36454544353000383409943910749, −1.33762887582947598767878424547, −1.30502716412589015946718347461, −1.12349504856397877668400114631, −1.10174323465576134481561842489, −1.07854877141477012459639214717, −0.953240475569587799575268923830, −0.946906606420737313287241053218, −0.904564921388826744152560727779, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.904564921388826744152560727779, 0.946906606420737313287241053218, 0.953240475569587799575268923830, 1.07854877141477012459639214717, 1.10174323465576134481561842489, 1.12349504856397877668400114631, 1.30502716412589015946718347461, 1.33762887582947598767878424547, 1.36454544353000383409943910749, 1.48755758827704879203405734711, 1.53927324392953859261830940732, 1.75749280291071720437399199608, 1.79019018887551384872374682982, 1.81329266254869614951347237511, 1.90626658867245593585975265043, 1.99997276896997845658011015985, 2.04804898273480102548309523599, 2.08391901213598401675018090560, 2.23749045348730158485048570554, 2.24537652181271127555958974599, 2.32246138668513310679957205749, 2.47297237950993398159363402502, 2.50387630881054782008760339597, 2.58090701112526895706712583342, 2.65623199979130886835599235353
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.