Dirichlet series
| L(s) = 1 | + 3·2-s + 2·3-s − 22·4-s − 200·5-s + 6·6-s − 24·8-s − 361·9-s − 600·10-s + 120·11-s − 44·12-s − 1.99e3·13-s − 400·15-s + 377·16-s − 1.85e3·17-s − 1.08e3·18-s + 1.82e3·19-s + 4.40e3·20-s + 360·22-s + 4.82e3·23-s − 48·24-s + 2.25e4·25-s − 5.98e3·26-s − 3.48e3·27-s + 1.05e4·29-s − 1.20e3·30-s − 5.14e3·31-s + 599·32-s + ⋯ |
| L(s) = 1 | + 0.530·2-s + 0.128·3-s − 0.687·4-s − 3.57·5-s + 0.0680·6-s − 0.132·8-s − 1.48·9-s − 1.89·10-s + 0.299·11-s − 0.0882·12-s − 3.27·13-s − 0.459·15-s + 0.368·16-s − 1.55·17-s − 0.787·18-s + 1.16·19-s + 2.45·20-s + 0.158·22-s + 1.90·23-s − 0.0170·24-s + 36/5·25-s − 1.73·26-s − 0.920·27-s + 2.31·29-s − 0.243·30-s − 0.962·31-s + 0.103·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{16}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(5^{8} \cdot 7^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.68346\times 10^{12}\) |
| Root analytic conductor: | \(6.26849\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 5^{8} \cdot 7^{16} ,\ ( \ : [5/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(1.679299487\) |
| \(L(\frac12)\) | \(\approx\) | \(1.679299487\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 + p^{2} T )^{8} \) |
| 7 | \( 1 \) | |
| good | 2 | \( 1 - 3 T + 31 T^{2} - 135 T^{3} + 319 p T^{4} - 451 p^{3} T^{5} + 5 p^{9} T^{6} + 7541 p^{4} T^{7} - 6113 p^{5} T^{8} + 7541 p^{9} T^{9} + 5 p^{19} T^{10} - 451 p^{18} T^{11} + 319 p^{21} T^{12} - 135 p^{25} T^{13} + 31 p^{30} T^{14} - 3 p^{35} T^{15} + p^{40} T^{16} \) |
| 3 | \( 1 - 2 T + 365 T^{2} + 226 p^{2} T^{3} + 34643 p T^{4} + 1978760 T^{5} + 8960390 p T^{6} + 6969620 p^{4} T^{7} + 254174510 p^{3} T^{8} + 6969620 p^{9} T^{9} + 8960390 p^{11} T^{10} + 1978760 p^{15} T^{11} + 34643 p^{21} T^{12} + 226 p^{27} T^{13} + 365 p^{30} T^{14} - 2 p^{35} T^{15} + p^{40} T^{16} \) | |
| 11 | \( 1 - 120 T + 650439 T^{2} - 8415556 p T^{3} + 206248742390 T^{4} - 31936634547460 T^{5} + 45502134837951841 T^{6} - 6934794188050192160 T^{7} + \)\(80\!\cdots\!78\)\( T^{8} - 6934794188050192160 p^{5} T^{9} + 45502134837951841 p^{10} T^{10} - 31936634547460 p^{15} T^{11} + 206248742390 p^{20} T^{12} - 8415556 p^{26} T^{13} + 650439 p^{30} T^{14} - 120 p^{35} T^{15} + p^{40} T^{16} \) | |
| 13 | \( 1 + 1994 T + 3552589 T^{2} + 4277883130 T^{3} + 4645686857590 T^{4} + 4122835412643502 T^{5} + 258211337946918791 p T^{6} + \)\(23\!\cdots\!98\)\( T^{7} + \)\(15\!\cdots\!18\)\( T^{8} + \)\(23\!\cdots\!98\)\( p^{5} T^{9} + 258211337946918791 p^{11} T^{10} + 4122835412643502 p^{15} T^{11} + 4645686857590 p^{20} T^{12} + 4277883130 p^{25} T^{13} + 3552589 p^{30} T^{14} + 1994 p^{35} T^{15} + p^{40} T^{16} \) | |
| 17 | \( 1 + 1856 T + 5422560 T^{2} + 4567754272 T^{3} + 7977429556524 T^{4} - 1268416086303200 T^{5} + 1571134632914973280 T^{6} - \)\(14\!\cdots\!60\)\( T^{7} - \)\(48\!\cdots\!30\)\( T^{8} - \)\(14\!\cdots\!60\)\( p^{5} T^{9} + 1571134632914973280 p^{10} T^{10} - 1268416086303200 p^{15} T^{11} + 7977429556524 p^{20} T^{12} + 4567754272 p^{25} T^{13} + 5422560 p^{30} T^{14} + 1856 p^{35} T^{15} + p^{40} T^{16} \) | |
| 19 | \( 1 - 1828 T + 8895091 T^{2} - 13615442468 T^{3} + 42883793499898 T^{4} - 67430200554256996 T^{5} + \)\(15\!\cdots\!57\)\( T^{6} - \)\(22\!\cdots\!40\)\( T^{7} + \)\(41\!\cdots\!74\)\( T^{8} - \)\(22\!\cdots\!40\)\( p^{5} T^{9} + \)\(15\!\cdots\!57\)\( p^{10} T^{10} - 67430200554256996 p^{15} T^{11} + 42883793499898 p^{20} T^{12} - 13615442468 p^{25} T^{13} + 8895091 p^{30} T^{14} - 1828 p^{35} T^{15} + p^{40} T^{16} \) | |
| 23 | \( 1 - 4822 T + 42288532 T^{2} - 157015754260 T^{3} + 765865376535214 T^{4} - 2283893923105986802 T^{5} + \)\(82\!\cdots\!00\)\( T^{6} - \)\(20\!\cdots\!14\)\( T^{7} + \)\(61\!\cdots\!55\)\( T^{8} - \)\(20\!\cdots\!14\)\( p^{5} T^{9} + \)\(82\!\cdots\!00\)\( p^{10} T^{10} - 2283893923105986802 p^{15} T^{11} + 765865376535214 p^{20} T^{12} - 157015754260 p^{25} T^{13} + 42288532 p^{30} T^{14} - 4822 p^{35} T^{15} + p^{40} T^{16} \) | |
| 29 | \( 1 - 10502 T + 126609859 T^{2} - 959906587162 T^{3} + 7575035215805425 T^{4} - 45501895918818387492 T^{5} + \)\(27\!\cdots\!78\)\( T^{6} - \)\(13\!\cdots\!00\)\( T^{7} + \)\(68\!\cdots\!74\)\( T^{8} - \)\(13\!\cdots\!00\)\( p^{5} T^{9} + \)\(27\!\cdots\!78\)\( p^{10} T^{10} - 45501895918818387492 p^{15} T^{11} + 7575035215805425 p^{20} T^{12} - 959906587162 p^{25} T^{13} + 126609859 p^{30} T^{14} - 10502 p^{35} T^{15} + p^{40} T^{16} \) | |
| 31 | \( 1 + 5148 T + 177783988 T^{2} + 939954268572 T^{3} + 15172607261694628 T^{4} + 74011848860784362412 T^{5} + \)\(80\!\cdots\!48\)\( T^{6} + \)\(33\!\cdots\!32\)\( T^{7} + \)\(29\!\cdots\!18\)\( p^{2} T^{8} + \)\(33\!\cdots\!32\)\( p^{5} T^{9} + \)\(80\!\cdots\!48\)\( p^{10} T^{10} + 74011848860784362412 p^{15} T^{11} + 15172607261694628 p^{20} T^{12} + 939954268572 p^{25} T^{13} + 177783988 p^{30} T^{14} + 5148 p^{35} T^{15} + p^{40} T^{16} \) | |
| 37 | \( 1 - 7810 T + 414360329 T^{2} - 3533395732230 T^{3} + 80485561430855034 T^{4} - \)\(69\!\cdots\!50\)\( T^{5} + \)\(97\!\cdots\!35\)\( T^{6} - \)\(21\!\cdots\!90\)\( p T^{7} + \)\(80\!\cdots\!50\)\( T^{8} - \)\(21\!\cdots\!90\)\( p^{6} T^{9} + \)\(97\!\cdots\!35\)\( p^{10} T^{10} - \)\(69\!\cdots\!50\)\( p^{15} T^{11} + 80485561430855034 p^{20} T^{12} - 3533395732230 p^{25} T^{13} + 414360329 p^{30} T^{14} - 7810 p^{35} T^{15} + p^{40} T^{16} \) | |
| 41 | \( 1 - 18192 T + 676061604 T^{2} - 10256472999296 T^{3} + 5286748626954058 p T^{4} - \)\(27\!\cdots\!68\)\( T^{5} + \)\(43\!\cdots\!44\)\( T^{6} - \)\(47\!\cdots\!48\)\( T^{7} + \)\(60\!\cdots\!63\)\( T^{8} - \)\(47\!\cdots\!48\)\( p^{5} T^{9} + \)\(43\!\cdots\!44\)\( p^{10} T^{10} - \)\(27\!\cdots\!68\)\( p^{15} T^{11} + 5286748626954058 p^{21} T^{12} - 10256472999296 p^{25} T^{13} + 676061604 p^{30} T^{14} - 18192 p^{35} T^{15} + p^{40} T^{16} \) | |
| 43 | \( 1 - 63190 T + 2648761301 T^{2} - 78856016900170 T^{3} + 1902206760906111209 T^{4} - \)\(37\!\cdots\!20\)\( T^{5} + \)\(64\!\cdots\!42\)\( T^{6} - \)\(95\!\cdots\!60\)\( T^{7} + \)\(12\!\cdots\!30\)\( T^{8} - \)\(95\!\cdots\!60\)\( p^{5} T^{9} + \)\(64\!\cdots\!42\)\( p^{10} T^{10} - \)\(37\!\cdots\!20\)\( p^{15} T^{11} + 1902206760906111209 p^{20} T^{12} - 78856016900170 p^{25} T^{13} + 2648761301 p^{30} T^{14} - 63190 p^{35} T^{15} + p^{40} T^{16} \) | |
| 47 | \( 1 + 11816 T + 1210232287 T^{2} + 14128959642716 T^{3} + 704962347840174486 T^{4} + \)\(79\!\cdots\!88\)\( T^{5} + \)\(26\!\cdots\!45\)\( T^{6} + \)\(27\!\cdots\!44\)\( T^{7} + \)\(70\!\cdots\!82\)\( T^{8} + \)\(27\!\cdots\!44\)\( p^{5} T^{9} + \)\(26\!\cdots\!45\)\( p^{10} T^{10} + \)\(79\!\cdots\!88\)\( p^{15} T^{11} + 704962347840174486 p^{20} T^{12} + 14128959642716 p^{25} T^{13} + 1210232287 p^{30} T^{14} + 11816 p^{35} T^{15} + p^{40} T^{16} \) | |
| 53 | \( 1 + 39902 T + 1233014409 T^{2} + 21938625043746 T^{3} + 547751988741547474 T^{4} + \)\(98\!\cdots\!70\)\( T^{5} + \)\(30\!\cdots\!95\)\( T^{6} + \)\(75\!\cdots\!90\)\( T^{7} + \)\(20\!\cdots\!90\)\( T^{8} + \)\(75\!\cdots\!90\)\( p^{5} T^{9} + \)\(30\!\cdots\!95\)\( p^{10} T^{10} + \)\(98\!\cdots\!70\)\( p^{15} T^{11} + 547751988741547474 p^{20} T^{12} + 21938625043746 p^{25} T^{13} + 1233014409 p^{30} T^{14} + 39902 p^{35} T^{15} + p^{40} T^{16} \) | |
| 59 | \( 1 + 50752 T + 3652630672 T^{2} + 141858008796288 T^{3} + 6140355307710080876 T^{4} + \)\(19\!\cdots\!20\)\( T^{5} + \)\(65\!\cdots\!48\)\( T^{6} + \)\(17\!\cdots\!36\)\( T^{7} + \)\(52\!\cdots\!06\)\( T^{8} + \)\(17\!\cdots\!36\)\( p^{5} T^{9} + \)\(65\!\cdots\!48\)\( p^{10} T^{10} + \)\(19\!\cdots\!20\)\( p^{15} T^{11} + 6140355307710080876 p^{20} T^{12} + 141858008796288 p^{25} T^{13} + 3652630672 p^{30} T^{14} + 50752 p^{35} T^{15} + p^{40} T^{16} \) | |
| 61 | \( 1 - 2146 T + 4380250439 T^{2} + 14364535044078 T^{3} + 9031064828045619837 T^{4} + \)\(63\!\cdots\!56\)\( T^{5} + \)\(12\!\cdots\!82\)\( T^{6} + \)\(10\!\cdots\!88\)\( T^{7} + \)\(11\!\cdots\!34\)\( T^{8} + \)\(10\!\cdots\!88\)\( p^{5} T^{9} + \)\(12\!\cdots\!82\)\( p^{10} T^{10} + \)\(63\!\cdots\!56\)\( p^{15} T^{11} + 9031064828045619837 p^{20} T^{12} + 14364535044078 p^{25} T^{13} + 4380250439 p^{30} T^{14} - 2146 p^{35} T^{15} + p^{40} T^{16} \) | |
| 67 | \( 1 - 50498 T + 8999506137 T^{2} - 341475252270238 T^{3} + 35276062732380361461 T^{4} - \)\(10\!\cdots\!04\)\( T^{5} + \)\(83\!\cdots\!50\)\( T^{6} - \)\(21\!\cdots\!72\)\( T^{7} + \)\(13\!\cdots\!02\)\( T^{8} - \)\(21\!\cdots\!72\)\( p^{5} T^{9} + \)\(83\!\cdots\!50\)\( p^{10} T^{10} - \)\(10\!\cdots\!04\)\( p^{15} T^{11} + 35276062732380361461 p^{20} T^{12} - 341475252270238 p^{25} T^{13} + 8999506137 p^{30} T^{14} - 50498 p^{35} T^{15} + p^{40} T^{16} \) | |
| 71 | \( 1 - 183976 T + 26400248496 T^{2} - 2617247925179688 T^{3} + \)\(22\!\cdots\!40\)\( T^{4} - \)\(15\!\cdots\!80\)\( T^{5} + \)\(91\!\cdots\!60\)\( T^{6} - \)\(47\!\cdots\!56\)\( T^{7} + \)\(21\!\cdots\!06\)\( T^{8} - \)\(47\!\cdots\!56\)\( p^{5} T^{9} + \)\(91\!\cdots\!60\)\( p^{10} T^{10} - \)\(15\!\cdots\!80\)\( p^{15} T^{11} + \)\(22\!\cdots\!40\)\( p^{20} T^{12} - 2617247925179688 p^{25} T^{13} + 26400248496 p^{30} T^{14} - 183976 p^{35} T^{15} + p^{40} T^{16} \) | |
| 73 | \( 1 + 54436 T + 8941200572 T^{2} + 484972061488348 T^{3} + 40093425746201282564 T^{4} + \)\(19\!\cdots\!00\)\( T^{5} + \)\(12\!\cdots\!40\)\( T^{6} + \)\(53\!\cdots\!40\)\( T^{7} + \)\(29\!\cdots\!50\)\( T^{8} + \)\(53\!\cdots\!40\)\( p^{5} T^{9} + \)\(12\!\cdots\!40\)\( p^{10} T^{10} + \)\(19\!\cdots\!00\)\( p^{15} T^{11} + 40093425746201282564 p^{20} T^{12} + 484972061488348 p^{25} T^{13} + 8941200572 p^{30} T^{14} + 54436 p^{35} T^{15} + p^{40} T^{16} \) | |
| 79 | \( 1 - 51040 T + 15114797372 T^{2} - 851357292497472 T^{3} + 1400358766991247580 p T^{4} - \)\(62\!\cdots\!72\)\( T^{5} + \)\(53\!\cdots\!16\)\( T^{6} - \)\(27\!\cdots\!32\)\( T^{7} + \)\(19\!\cdots\!18\)\( T^{8} - \)\(27\!\cdots\!32\)\( p^{5} T^{9} + \)\(53\!\cdots\!16\)\( p^{10} T^{10} - \)\(62\!\cdots\!72\)\( p^{15} T^{11} + 1400358766991247580 p^{21} T^{12} - 851357292497472 p^{25} T^{13} + 15114797372 p^{30} T^{14} - 51040 p^{35} T^{15} + p^{40} T^{16} \) | |
| 83 | \( 1 + 60438 T + 25504109265 T^{2} + 1154106312774874 T^{3} + \)\(29\!\cdots\!01\)\( T^{4} + \)\(10\!\cdots\!56\)\( T^{5} + \)\(20\!\cdots\!30\)\( T^{6} + \)\(61\!\cdots\!68\)\( T^{7} + \)\(97\!\cdots\!46\)\( T^{8} + \)\(61\!\cdots\!68\)\( p^{5} T^{9} + \)\(20\!\cdots\!30\)\( p^{10} T^{10} + \)\(10\!\cdots\!56\)\( p^{15} T^{11} + \)\(29\!\cdots\!01\)\( p^{20} T^{12} + 1154106312774874 p^{25} T^{13} + 25504109265 p^{30} T^{14} + 60438 p^{35} T^{15} + p^{40} T^{16} \) | |
| 89 | \( 1 - 96678 T + 34939392603 T^{2} - 2431783842365322 T^{3} + \)\(50\!\cdots\!13\)\( T^{4} - \)\(25\!\cdots\!20\)\( T^{5} + \)\(43\!\cdots\!82\)\( T^{6} - \)\(16\!\cdots\!44\)\( T^{7} + \)\(26\!\cdots\!90\)\( T^{8} - \)\(16\!\cdots\!44\)\( p^{5} T^{9} + \)\(43\!\cdots\!82\)\( p^{10} T^{10} - \)\(25\!\cdots\!20\)\( p^{15} T^{11} + \)\(50\!\cdots\!13\)\( p^{20} T^{12} - 2431783842365322 p^{25} T^{13} + 34939392603 p^{30} T^{14} - 96678 p^{35} T^{15} + p^{40} T^{16} \) | |
| 97 | \( 1 - 195312 T + 51464827096 T^{2} - 7592883349187920 T^{3} + \)\(12\!\cdots\!20\)\( T^{4} - \)\(15\!\cdots\!76\)\( T^{5} + \)\(19\!\cdots\!32\)\( T^{6} - \)\(19\!\cdots\!96\)\( T^{7} + \)\(19\!\cdots\!58\)\( T^{8} - \)\(19\!\cdots\!96\)\( p^{5} T^{9} + \)\(19\!\cdots\!32\)\( p^{10} T^{10} - \)\(15\!\cdots\!76\)\( p^{15} T^{11} + \)\(12\!\cdots\!20\)\( p^{20} T^{12} - 7592883349187920 p^{25} T^{13} + 51464827096 p^{30} T^{14} - 195312 p^{35} T^{15} + p^{40} T^{16} \) | |
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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
−4.60726423679691831073619461850, −4.19829469177704379976111738371, −4.15698405654729641985932855083, −4.14230695602383856323732593400, −3.97750141319267285479165062893, −3.74600893608931983879658276785, −3.53345266731183708379358188927, −3.51806482888419343949205601495, −3.01767244881553936621142715511, −3.00427803570023987068097500536, −2.89073418380423223982098965376, −2.86313216050129286802573859413, −2.75012438837752704011232105790, −2.36957649279786259782977440387, −2.17976841565124299895564697216, −2.17964870811654006540162037123, −1.90503948069574463657394291788, −1.50022666311801853086406147970, −1.00641748025717431402131588676, −0.905583554050149278598374000749, −0.904295701342975438448640401015, −0.62718166316024071034501783424, −0.44227234275711349885657814871, −0.38554831149913700129383917520, −0.15501593004178051487544290783, 0.15501593004178051487544290783, 0.38554831149913700129383917520, 0.44227234275711349885657814871, 0.62718166316024071034501783424, 0.904295701342975438448640401015, 0.905583554050149278598374000749, 1.00641748025717431402131588676, 1.50022666311801853086406147970, 1.90503948069574463657394291788, 2.17964870811654006540162037123, 2.17976841565124299895564697216, 2.36957649279786259782977440387, 2.75012438837752704011232105790, 2.86313216050129286802573859413, 2.89073418380423223982098965376, 3.00427803570023987068097500536, 3.01767244881553936621142715511, 3.51806482888419343949205601495, 3.53345266731183708379358188927, 3.74600893608931983879658276785, 3.97750141319267285479165062893, 4.14230695602383856323732593400, 4.15698405654729641985932855083, 4.19829469177704379976111738371, 4.60726423679691831073619461850
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.