Dirichlet series
| L(s) = 1 | + 12·2-s − 9·3-s + 60·4-s − 36·5-s − 108·6-s + 25·7-s + 112·8-s + 65·9-s − 432·10-s + 37·11-s − 540·12-s + 300·14-s + 324·15-s − 336·16-s − 99·17-s + 780·18-s − 81·19-s − 2.16e3·20-s − 225·21-s + 444·22-s − 267·23-s − 1.00e3·24-s + 266·25-s − 58·27-s + 1.50e3·28-s + 119·29-s + 3.88e3·30-s + ⋯ |
| L(s) = 1 | + 4.24·2-s − 1.73·3-s + 15/2·4-s − 3.21·5-s − 7.34·6-s + 1.34·7-s + 4.94·8-s + 2.40·9-s − 13.6·10-s + 1.01·11-s − 12.9·12-s + 5.72·14-s + 5.57·15-s − 5.25·16-s − 1.41·17-s + 10.2·18-s − 0.978·19-s − 24.1·20-s − 2.33·21-s + 4.30·22-s − 2.42·23-s − 8.57·24-s + 2.12·25-s − 0.413·27-s + 10.1·28-s + 0.761·29-s + 23.6·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 13^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.95724\times 10^{15}\) |
| Root analytic conductor: | \(4.46571\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 13^{24} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.8583291842\) |
| \(L(\frac12)\) | \(\approx\) | \(0.8583291842\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T + p^{2} T^{2} )^{6} \) |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p^{2} T + 16 T^{2} - 383 T^{3} - 2903 T^{4} - 1330 p T^{5} + 9848 p^{2} T^{6} + 594275 T^{7} + 114521 p T^{8} - 15120890 T^{9} - 65767805 T^{10} + 130939027 T^{11} + 2123613139 T^{12} + 130939027 p^{3} T^{13} - 65767805 p^{6} T^{14} - 15120890 p^{9} T^{15} + 114521 p^{13} T^{16} + 594275 p^{15} T^{17} + 9848 p^{20} T^{18} - 1330 p^{22} T^{19} - 2903 p^{24} T^{20} - 383 p^{27} T^{21} + 16 p^{30} T^{22} + p^{35} T^{23} + p^{36} T^{24} \) |
| 5 | \( ( 1 + 18 T + 353 T^{2} + 1063 p T^{3} + 15993 p T^{4} + 1003083 T^{5} + 12801634 T^{6} + 1003083 p^{3} T^{7} + 15993 p^{7} T^{8} + 1063 p^{10} T^{9} + 353 p^{12} T^{10} + 18 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 7 | \( 1 - 25 T - 267 T^{2} + 1544 p T^{3} - 41021 T^{4} - 1296635 T^{5} + 23995119 T^{6} + 9910501 p T^{7} - 1232685738 p T^{8} - 31346785682 p T^{9} + 154528034457 p^{2} T^{10} + 1657960494149 p^{2} T^{11} - 89824498888557 p^{2} T^{12} + 1657960494149 p^{5} T^{13} + 154528034457 p^{8} T^{14} - 31346785682 p^{10} T^{15} - 1232685738 p^{13} T^{16} + 9910501 p^{16} T^{17} + 23995119 p^{18} T^{18} - 1296635 p^{21} T^{19} - 41021 p^{24} T^{20} + 1544 p^{28} T^{21} - 267 p^{30} T^{22} - 25 p^{33} T^{23} + p^{36} T^{24} \) | |
| 11 | \( 1 - 37 T - 3374 T^{2} + 90165 T^{3} + 6920025 T^{4} - 72282084 T^{5} - 9465779646 T^{6} - 122761204227 T^{7} + 992732213841 p T^{8} + 2107273243932 p^{2} T^{9} - 5376348087443175 T^{10} - 186152687993983949 T^{11} + 1687873828970897175 T^{12} - 186152687993983949 p^{3} T^{13} - 5376348087443175 p^{6} T^{14} + 2107273243932 p^{11} T^{15} + 992732213841 p^{13} T^{16} - 122761204227 p^{15} T^{17} - 9465779646 p^{18} T^{18} - 72282084 p^{21} T^{19} + 6920025 p^{24} T^{20} + 90165 p^{27} T^{21} - 3374 p^{30} T^{22} - 37 p^{33} T^{23} + p^{36} T^{24} \) | |
| 17 | \( 1 + 99 T - 5434 T^{2} - 1327755 T^{3} - 64109215 T^{4} + 2302353130 T^{5} + 284397483536 T^{6} + 14376654383487 T^{7} + 1730888463481815 T^{8} + 160829292478556350 T^{9} + 2491616674911046785 T^{10} - \)\(92\!\cdots\!33\)\( T^{11} - \)\(97\!\cdots\!91\)\( T^{12} - \)\(92\!\cdots\!33\)\( p^{3} T^{13} + 2491616674911046785 p^{6} T^{14} + 160829292478556350 p^{9} T^{15} + 1730888463481815 p^{12} T^{16} + 14376654383487 p^{15} T^{17} + 284397483536 p^{18} T^{18} + 2302353130 p^{21} T^{19} - 64109215 p^{24} T^{20} - 1327755 p^{27} T^{21} - 5434 p^{30} T^{22} + 99 p^{33} T^{23} + p^{36} T^{24} \) | |
| 19 | \( 1 + 81 T - 14726 T^{2} - 186209 T^{3} + 166884549 T^{4} - 6779420944 T^{5} - 866610386870 T^{6} + 89433153834823 T^{7} + 664547746074283 T^{8} - 502153099717971364 T^{9} + 11987237177864360773 T^{10} + 56646495765690815391 p T^{11} - \)\(88\!\cdots\!53\)\( T^{12} + 56646495765690815391 p^{4} T^{13} + 11987237177864360773 p^{6} T^{14} - 502153099717971364 p^{9} T^{15} + 664547746074283 p^{12} T^{16} + 89433153834823 p^{15} T^{17} - 866610386870 p^{18} T^{18} - 6779420944 p^{21} T^{19} + 166884549 p^{24} T^{20} - 186209 p^{27} T^{21} - 14726 p^{30} T^{22} + 81 p^{33} T^{23} + p^{36} T^{24} \) | |
| 23 | \( 1 + 267 T + 5996 T^{2} - 5270773 T^{3} - 672443396 T^{4} - 31932625905 T^{5} - 1227012468944 T^{6} + 664024075258251 T^{7} + 222364660120074076 T^{8} + 18529838508574461255 T^{9} - \)\(82\!\cdots\!32\)\( T^{10} - \)\(95\!\cdots\!35\)\( p T^{11} - \)\(20\!\cdots\!18\)\( T^{12} - \)\(95\!\cdots\!35\)\( p^{4} T^{13} - \)\(82\!\cdots\!32\)\( p^{6} T^{14} + 18529838508574461255 p^{9} T^{15} + 222364660120074076 p^{12} T^{16} + 664024075258251 p^{15} T^{17} - 1227012468944 p^{18} T^{18} - 31932625905 p^{21} T^{19} - 672443396 p^{24} T^{20} - 5270773 p^{27} T^{21} + 5996 p^{30} T^{22} + 267 p^{33} T^{23} + p^{36} T^{24} \) | |
| 29 | \( 1 - 119 T - 110183 T^{2} + 8876420 T^{3} + 7293704953 T^{4} - 379237642413 T^{5} - 340028891349473 T^{6} + 10660672822726647 T^{7} + 12332118877300770900 T^{8} - \)\(19\!\cdots\!14\)\( T^{9} - \)\(37\!\cdots\!75\)\( T^{10} + \)\(63\!\cdots\!69\)\( p T^{11} + \)\(11\!\cdots\!75\)\( p^{2} T^{12} + \)\(63\!\cdots\!69\)\( p^{4} T^{13} - \)\(37\!\cdots\!75\)\( p^{6} T^{14} - \)\(19\!\cdots\!14\)\( p^{9} T^{15} + 12332118877300770900 p^{12} T^{16} + 10660672822726647 p^{15} T^{17} - 340028891349473 p^{18} T^{18} - 379237642413 p^{21} T^{19} + 7293704953 p^{24} T^{20} + 8876420 p^{27} T^{21} - 110183 p^{30} T^{22} - 119 p^{33} T^{23} + p^{36} T^{24} \) | |
| 31 | \( ( 1 - 625 T + 316140 T^{2} - 104356466 T^{3} + 29721828071 T^{4} - 6530570693913 T^{5} + 1262869136265616 T^{6} - 6530570693913 p^{3} T^{7} + 29721828071 p^{6} T^{8} - 104356466 p^{9} T^{9} + 316140 p^{12} T^{10} - 625 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 37 | \( 1 - 274 T - 172179 T^{2} + 31298256 T^{3} + 20598937694 T^{4} - 1795401545956 T^{5} - 1915135012258020 T^{6} + 108121656871938888 T^{7} + \)\(13\!\cdots\!13\)\( T^{8} - \)\(43\!\cdots\!40\)\( T^{9} - \)\(83\!\cdots\!93\)\( T^{10} + \)\(63\!\cdots\!30\)\( T^{11} + \)\(46\!\cdots\!61\)\( T^{12} + \)\(63\!\cdots\!30\)\( p^{3} T^{13} - \)\(83\!\cdots\!93\)\( p^{6} T^{14} - \)\(43\!\cdots\!40\)\( p^{9} T^{15} + \)\(13\!\cdots\!13\)\( p^{12} T^{16} + 108121656871938888 p^{15} T^{17} - 1915135012258020 p^{18} T^{18} - 1795401545956 p^{21} T^{19} + 20598937694 p^{24} T^{20} + 31298256 p^{27} T^{21} - 172179 p^{30} T^{22} - 274 p^{33} T^{23} + p^{36} T^{24} \) | |
| 41 | \( 1 - 1140 T + 431902 T^{2} - 53291206 T^{3} + 18188754925 T^{4} - 16860355235744 T^{5} + 5182958888100297 T^{6} - 1001584729781787693 T^{7} + 9902716607030270666 p T^{8} - \)\(12\!\cdots\!62\)\( T^{9} + \)\(19\!\cdots\!99\)\( T^{10} - \)\(81\!\cdots\!15\)\( T^{11} + \)\(33\!\cdots\!16\)\( T^{12} - \)\(81\!\cdots\!15\)\( p^{3} T^{13} + \)\(19\!\cdots\!99\)\( p^{6} T^{14} - \)\(12\!\cdots\!62\)\( p^{9} T^{15} + 9902716607030270666 p^{13} T^{16} - 1001584729781787693 p^{15} T^{17} + 5182958888100297 p^{18} T^{18} - 16860355235744 p^{21} T^{19} + 18188754925 p^{24} T^{20} - 53291206 p^{27} T^{21} + 431902 p^{30} T^{22} - 1140 p^{33} T^{23} + p^{36} T^{24} \) | |
| 43 | \( 1 + 428 T - 16653 T^{2} - 65902532 T^{3} - 28639656928 T^{4} - 5977021820724 T^{5} + 249322836785029 T^{6} + 859861367196461312 T^{7} + \)\(40\!\cdots\!87\)\( T^{8} + \)\(83\!\cdots\!00\)\( T^{9} - \)\(29\!\cdots\!48\)\( T^{10} - \)\(78\!\cdots\!64\)\( T^{11} - \)\(30\!\cdots\!75\)\( T^{12} - \)\(78\!\cdots\!64\)\( p^{3} T^{13} - \)\(29\!\cdots\!48\)\( p^{6} T^{14} + \)\(83\!\cdots\!00\)\( p^{9} T^{15} + \)\(40\!\cdots\!87\)\( p^{12} T^{16} + 859861367196461312 p^{15} T^{17} + 249322836785029 p^{18} T^{18} - 5977021820724 p^{21} T^{19} - 28639656928 p^{24} T^{20} - 65902532 p^{27} T^{21} - 16653 p^{30} T^{22} + 428 p^{33} T^{23} + p^{36} T^{24} \) | |
| 47 | \( ( 1 - 986 T + 712667 T^{2} - 325045705 T^{3} + 134208823841 T^{4} - 44205691637095 T^{5} + 15291628070110878 T^{6} - 44205691637095 p^{3} T^{7} + 134208823841 p^{6} T^{8} - 325045705 p^{9} T^{9} + 712667 p^{12} T^{10} - 986 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 53 | \( ( 1 - 89 T + 149636 T^{2} + 19578912 T^{3} + 11883866543 T^{4} - 8498293720903 T^{5} + 5004078025030288 T^{6} - 8498293720903 p^{3} T^{7} + 11883866543 p^{6} T^{8} + 19578912 p^{9} T^{9} + 149636 p^{12} T^{10} - 89 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 59 | \( 1 - 1088 T + 328780 T^{2} + 90009146 T^{3} - 84336779359 T^{4} + 13283513005596 T^{5} + 21739034235208103 T^{6} - 17322876409308491411 T^{7} + \)\(17\!\cdots\!02\)\( T^{8} + \)\(27\!\cdots\!32\)\( T^{9} - \)\(61\!\cdots\!71\)\( T^{10} - \)\(47\!\cdots\!91\)\( T^{11} + \)\(35\!\cdots\!96\)\( T^{12} - \)\(47\!\cdots\!91\)\( p^{3} T^{13} - \)\(61\!\cdots\!71\)\( p^{6} T^{14} + \)\(27\!\cdots\!32\)\( p^{9} T^{15} + \)\(17\!\cdots\!02\)\( p^{12} T^{16} - 17322876409308491411 p^{15} T^{17} + 21739034235208103 p^{18} T^{18} + 13283513005596 p^{21} T^{19} - 84336779359 p^{24} T^{20} + 90009146 p^{27} T^{21} + 328780 p^{30} T^{22} - 1088 p^{33} T^{23} + p^{36} T^{24} \) | |
| 61 | \( 1 + 1704 T + 964913 T^{2} + 32814990 T^{3} - 190132450110 T^{4} - 105404733375058 T^{5} - 40545666342004564 T^{6} + 2060628600521217352 T^{7} + \)\(18\!\cdots\!69\)\( T^{8} + \)\(88\!\cdots\!24\)\( T^{9} - \)\(33\!\cdots\!05\)\( T^{10} - \)\(16\!\cdots\!30\)\( T^{11} - \)\(87\!\cdots\!63\)\( T^{12} - \)\(16\!\cdots\!30\)\( p^{3} T^{13} - \)\(33\!\cdots\!05\)\( p^{6} T^{14} + \)\(88\!\cdots\!24\)\( p^{9} T^{15} + \)\(18\!\cdots\!69\)\( p^{12} T^{16} + 2060628600521217352 p^{15} T^{17} - 40545666342004564 p^{18} T^{18} - 105404733375058 p^{21} T^{19} - 190132450110 p^{24} T^{20} + 32814990 p^{27} T^{21} + 964913 p^{30} T^{22} + 1704 p^{33} T^{23} + p^{36} T^{24} \) | |
| 67 | \( 1 + 1692 T + 127351 T^{2} - 843881488 T^{3} + 362460636146 T^{4} + 759339717368988 T^{5} - 94435239939881877 T^{6} - \)\(20\!\cdots\!64\)\( T^{7} + \)\(15\!\cdots\!87\)\( T^{8} + \)\(89\!\cdots\!64\)\( T^{9} - \)\(49\!\cdots\!48\)\( T^{10} - \)\(19\!\cdots\!86\)\( T^{11} + \)\(25\!\cdots\!73\)\( T^{12} - \)\(19\!\cdots\!86\)\( p^{3} T^{13} - \)\(49\!\cdots\!48\)\( p^{6} T^{14} + \)\(89\!\cdots\!64\)\( p^{9} T^{15} + \)\(15\!\cdots\!87\)\( p^{12} T^{16} - \)\(20\!\cdots\!64\)\( p^{15} T^{17} - 94435239939881877 p^{18} T^{18} + 759339717368988 p^{21} T^{19} + 362460636146 p^{24} T^{20} - 843881488 p^{27} T^{21} + 127351 p^{30} T^{22} + 1692 p^{33} T^{23} + p^{36} T^{24} \) | |
| 71 | \( 1 + 1221 T - 222229 T^{2} - 229445802 T^{3} + 548429440873 T^{4} + 98641092965829 T^{5} - 178758364328736769 T^{6} + 98487688450745149485 T^{7} + \)\(64\!\cdots\!56\)\( T^{8} - \)\(32\!\cdots\!66\)\( T^{9} + \)\(14\!\cdots\!11\)\( T^{10} + \)\(13\!\cdots\!13\)\( T^{11} - \)\(24\!\cdots\!05\)\( T^{12} + \)\(13\!\cdots\!13\)\( p^{3} T^{13} + \)\(14\!\cdots\!11\)\( p^{6} T^{14} - \)\(32\!\cdots\!66\)\( p^{9} T^{15} + \)\(64\!\cdots\!56\)\( p^{12} T^{16} + 98487688450745149485 p^{15} T^{17} - 178758364328736769 p^{18} T^{18} + 98641092965829 p^{21} T^{19} + 548429440873 p^{24} T^{20} - 229445802 p^{27} T^{21} - 222229 p^{30} T^{22} + 1221 p^{33} T^{23} + p^{36} T^{24} \) | |
| 73 | \( ( 1 - 1554 T + 2675277 T^{2} - 2821078736 T^{3} + 2754872512643 T^{4} - 2120598772053916 T^{5} + 1451353422113036309 T^{6} - 2120598772053916 p^{3} T^{7} + 2754872512643 p^{6} T^{8} - 2821078736 p^{9} T^{9} + 2675277 p^{12} T^{10} - 1554 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 79 | \( ( 1 + 875 T + 2505744 T^{2} + 1612380868 T^{3} + 2680533156433 T^{4} + 1358001363523637 T^{5} + 1672684167171147060 T^{6} + 1358001363523637 p^{3} T^{7} + 2680533156433 p^{6} T^{8} + 1612380868 p^{9} T^{9} + 2505744 p^{12} T^{10} + 875 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 83 | \( ( 1 - 126 T + 1578473 T^{2} - 79217558 T^{3} + 1520212835457 T^{4} + 29472962128638 T^{5} + 980791484443862659 T^{6} + 29472962128638 p^{3} T^{7} + 1520212835457 p^{6} T^{8} - 79217558 p^{9} T^{9} + 1578473 p^{12} T^{10} - 126 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 89 | \( 1 + 374 T - 2168759 T^{2} - 167755306 T^{3} + 2757626355714 T^{4} - 328578956137010 T^{5} - 1822411905315756507 T^{6} + \)\(86\!\cdots\!84\)\( T^{7} + \)\(43\!\cdots\!35\)\( T^{8} - \)\(67\!\cdots\!06\)\( T^{9} + \)\(62\!\cdots\!08\)\( T^{10} + \)\(24\!\cdots\!06\)\( T^{11} - \)\(70\!\cdots\!11\)\( T^{12} + \)\(24\!\cdots\!06\)\( p^{3} T^{13} + \)\(62\!\cdots\!08\)\( p^{6} T^{14} - \)\(67\!\cdots\!06\)\( p^{9} T^{15} + \)\(43\!\cdots\!35\)\( p^{12} T^{16} + \)\(86\!\cdots\!84\)\( p^{15} T^{17} - 1822411905315756507 p^{18} T^{18} - 328578956137010 p^{21} T^{19} + 2757626355714 p^{24} T^{20} - 167755306 p^{27} T^{21} - 2168759 p^{30} T^{22} + 374 p^{33} T^{23} + p^{36} T^{24} \) | |
| 97 | \( 1 + 330 T - 4456453 T^{2} - 2135265822 T^{3} + 11088575098674 T^{4} + 5879616897388588 T^{5} - 18416879164626935101 T^{6} - \)\(94\!\cdots\!10\)\( T^{7} + \)\(23\!\cdots\!93\)\( T^{8} + \)\(91\!\cdots\!58\)\( T^{9} - \)\(24\!\cdots\!18\)\( T^{10} - \)\(36\!\cdots\!34\)\( T^{11} + \)\(22\!\cdots\!29\)\( T^{12} - \)\(36\!\cdots\!34\)\( p^{3} T^{13} - \)\(24\!\cdots\!18\)\( p^{6} T^{14} + \)\(91\!\cdots\!58\)\( p^{9} T^{15} + \)\(23\!\cdots\!93\)\( p^{12} T^{16} - \)\(94\!\cdots\!10\)\( p^{15} T^{17} - 18416879164626935101 p^{18} T^{18} + 5879616897388588 p^{21} T^{19} + 11088575098674 p^{24} T^{20} - 2135265822 p^{27} T^{21} - 4456453 p^{30} T^{22} + 330 p^{33} T^{23} + p^{36} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.53237886113411524340728771355, −3.37679876248029380780868077252, −3.29967630988982248644577459690, −3.24233964973641692981141942281, −3.08443732134004839722248129114, −2.81992668026463742374165397079, −2.72526529883301546338933083387, −2.64352631567657573806976756859, −2.61336623161907359077961984679, −2.57762586093683013153948980981, −2.32778708963850150562451539333, −2.23225478123826717330491205753, −2.01929257029529925835906499601, −1.99898084331206432604088252998, −1.70580105612999719203534041685, −1.67998580616428711872636420873, −1.33985306769273469520210488402, −0.903329723735886571259340563529, −0.882273031208986115956641445603, −0.858427081618626560424266431864, −0.75887645925424578169659197639, −0.63667427378560946888801301842, −0.56684348546714581482050626667, −0.48604128536228963792787890561, −0.02068279098380190777965778585, 0.02068279098380190777965778585, 0.48604128536228963792787890561, 0.56684348546714581482050626667, 0.63667427378560946888801301842, 0.75887645925424578169659197639, 0.858427081618626560424266431864, 0.882273031208986115956641445603, 0.903329723735886571259340563529, 1.33985306769273469520210488402, 1.67998580616428711872636420873, 1.70580105612999719203534041685, 1.99898084331206432604088252998, 2.01929257029529925835906499601, 2.23225478123826717330491205753, 2.32778708963850150562451539333, 2.57762586093683013153948980981, 2.61336623161907359077961984679, 2.64352631567657573806976756859, 2.72526529883301546338933083387, 2.81992668026463742374165397079, 3.08443732134004839722248129114, 3.24233964973641692981141942281, 3.29967630988982248644577459690, 3.37679876248029380780868077252, 3.53237886113411524340728771355
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.