Dirichlet series
| L(s) = 1 | + 5·3-s − 2·5-s + 10·7-s + 4·9-s + 14·11-s − 9·13-s − 10·15-s + 4·17-s + 9·19-s + 50·21-s + 15·23-s − 19·25-s − 20·27-s + 17·29-s + 11·31-s + 70·33-s − 20·35-s − 29·37-s − 45·39-s + 14·41-s + 15·43-s − 8·45-s + 17·47-s + 18·49-s + 20·51-s − 7·53-s − 28·55-s + ⋯ |
| L(s) = 1 | + 2.88·3-s − 0.894·5-s + 3.77·7-s + 4/3·9-s + 4.22·11-s − 2.49·13-s − 2.58·15-s + 0.970·17-s + 2.06·19-s + 10.9·21-s + 3.12·23-s − 3.79·25-s − 3.84·27-s + 3.15·29-s + 1.97·31-s + 12.1·33-s − 3.38·35-s − 4.76·37-s − 7.20·39-s + 2.18·41-s + 2.28·43-s − 1.19·45-s + 2.47·47-s + 18/7·49-s + 2.80·51-s − 0.961·53-s − 3.77·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 167^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36} \cdot 167^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{36} \cdot 167^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.17271\times 10^{12}\) |
| Root analytic conductor: | \(3.26619\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{36} \cdot 167^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(189.3428226\) |
| \(L(\frac12)\) | \(\approx\) | \(189.3428226\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 167 | \( ( 1 - T )^{12} \) | |
| good | 3 | \( 1 - 5 T + 7 p T^{2} - 65 T^{3} + 20 p^{2} T^{4} - 416 T^{5} + 286 p T^{6} - 1532 T^{7} + 2456 T^{8} - 3338 T^{9} + 4105 T^{10} - 166 p^{3} T^{11} + 6566 T^{12} - 166 p^{4} T^{13} + 4105 p^{2} T^{14} - 3338 p^{3} T^{15} + 2456 p^{4} T^{16} - 1532 p^{5} T^{17} + 286 p^{7} T^{18} - 416 p^{7} T^{19} + 20 p^{10} T^{20} - 65 p^{9} T^{21} + 7 p^{11} T^{22} - 5 p^{11} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 + 2 T + 23 T^{2} + 48 T^{3} + 303 T^{4} + 618 T^{5} + 3004 T^{6} + 5842 T^{7} + 4753 p T^{8} + 43696 T^{9} + 30937 p T^{10} + 262994 T^{11} + 843222 T^{12} + 262994 p T^{13} + 30937 p^{3} T^{14} + 43696 p^{3} T^{15} + 4753 p^{5} T^{16} + 5842 p^{5} T^{17} + 3004 p^{6} T^{18} + 618 p^{7} T^{19} + 303 p^{8} T^{20} + 48 p^{9} T^{21} + 23 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 - 10 T + 82 T^{2} - 68 p T^{3} + 2479 T^{4} - 10943 T^{5} + 44794 T^{6} - 164522 T^{7} + 81359 p T^{8} - 1817643 T^{9} + 5531118 T^{10} - 15745437 T^{11} + 6139950 p T^{12} - 15745437 p T^{13} + 5531118 p^{2} T^{14} - 1817643 p^{3} T^{15} + 81359 p^{5} T^{16} - 164522 p^{5} T^{17} + 44794 p^{6} T^{18} - 10943 p^{7} T^{19} + 2479 p^{8} T^{20} - 68 p^{10} T^{21} + 82 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 - 14 T + 160 T^{2} - 1234 T^{3} + 762 p T^{4} - 46483 T^{5} + 238446 T^{6} - 1078238 T^{7} + 4676555 T^{8} - 18581954 T^{9} + 71789378 T^{10} - 256135639 T^{11} + 884942924 T^{12} - 256135639 p T^{13} + 71789378 p^{2} T^{14} - 18581954 p^{3} T^{15} + 4676555 p^{4} T^{16} - 1078238 p^{5} T^{17} + 238446 p^{6} T^{18} - 46483 p^{7} T^{19} + 762 p^{9} T^{20} - 1234 p^{9} T^{21} + 160 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 9 T + 124 T^{2} + 784 T^{3} + 6135 T^{4} + 29412 T^{5} + 166800 T^{6} + 624997 T^{7} + 2854483 T^{8} + 8529550 T^{9} + 2710404 p T^{10} + 92130964 T^{11} + 416218410 T^{12} + 92130964 p T^{13} + 2710404 p^{3} T^{14} + 8529550 p^{3} T^{15} + 2854483 p^{4} T^{16} + 624997 p^{5} T^{17} + 166800 p^{6} T^{18} + 29412 p^{7} T^{19} + 6135 p^{8} T^{20} + 784 p^{9} T^{21} + 124 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 4 T + 97 T^{2} - 264 T^{3} + 4341 T^{4} - 7296 T^{5} + 129101 T^{6} - 117284 T^{7} + 3130755 T^{8} - 1704552 T^{9} + 67580850 T^{10} - 32335528 T^{11} + 74036142 p T^{12} - 32335528 p T^{13} + 67580850 p^{2} T^{14} - 1704552 p^{3} T^{15} + 3130755 p^{4} T^{16} - 117284 p^{5} T^{17} + 129101 p^{6} T^{18} - 7296 p^{7} T^{19} + 4341 p^{8} T^{20} - 264 p^{9} T^{21} + 97 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 9 T + 127 T^{2} - 999 T^{3} + 8513 T^{4} - 55475 T^{5} + 374218 T^{6} - 2092084 T^{7} + 11981297 T^{8} - 59771665 T^{9} + 302921079 T^{10} - 71970978 p T^{11} + 6310019386 T^{12} - 71970978 p^{2} T^{13} + 302921079 p^{2} T^{14} - 59771665 p^{3} T^{15} + 11981297 p^{4} T^{16} - 2092084 p^{5} T^{17} + 374218 p^{6} T^{18} - 55475 p^{7} T^{19} + 8513 p^{8} T^{20} - 999 p^{9} T^{21} + 127 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 15 T + 268 T^{2} - 126 p T^{3} + 30599 T^{4} - 259246 T^{5} + 89292 p T^{6} - 14360493 T^{7} + 93286443 T^{8} - 558477374 T^{9} + 3118163168 T^{10} - 16394322468 T^{11} + 80741669146 T^{12} - 16394322468 p T^{13} + 3118163168 p^{2} T^{14} - 558477374 p^{3} T^{15} + 93286443 p^{4} T^{16} - 14360493 p^{5} T^{17} + 89292 p^{7} T^{18} - 259246 p^{7} T^{19} + 30599 p^{8} T^{20} - 126 p^{10} T^{21} + 268 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 17 T + 308 T^{2} - 3084 T^{3} + 32784 T^{4} - 239556 T^{5} + 1971086 T^{6} - 12005398 T^{7} + 88342328 T^{8} - 17012152 p T^{9} + 3403529628 T^{10} - 17607784369 T^{11} + 110114806362 T^{12} - 17607784369 p T^{13} + 3403529628 p^{2} T^{14} - 17012152 p^{4} T^{15} + 88342328 p^{4} T^{16} - 12005398 p^{5} T^{17} + 1971086 p^{6} T^{18} - 239556 p^{7} T^{19} + 32784 p^{8} T^{20} - 3084 p^{9} T^{21} + 308 p^{10} T^{22} - 17 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 11 T + 268 T^{2} - 2246 T^{3} + 31422 T^{4} - 208276 T^{5} + 2177752 T^{6} - 11625436 T^{7} + 102606128 T^{8} - 450844454 T^{9} + 3710634278 T^{10} - 14293752563 T^{11} + 118137620686 T^{12} - 14293752563 p T^{13} + 3710634278 p^{2} T^{14} - 450844454 p^{3} T^{15} + 102606128 p^{4} T^{16} - 11625436 p^{5} T^{17} + 2177752 p^{6} T^{18} - 208276 p^{7} T^{19} + 31422 p^{8} T^{20} - 2246 p^{9} T^{21} + 268 p^{10} T^{22} - 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 29 T + 631 T^{2} + 9945 T^{3} + 134321 T^{4} + 1544182 T^{5} + 15965302 T^{6} + 4000524 p T^{7} + 1262977605 T^{8} + 9884449483 T^{9} + 72006725027 T^{10} + 485946774853 T^{11} + 3066780197650 T^{12} + 485946774853 p T^{13} + 72006725027 p^{2} T^{14} + 9884449483 p^{3} T^{15} + 1262977605 p^{4} T^{16} + 4000524 p^{6} T^{17} + 15965302 p^{6} T^{18} + 1544182 p^{7} T^{19} + 134321 p^{8} T^{20} + 9945 p^{9} T^{21} + 631 p^{10} T^{22} + 29 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 14 T + 330 T^{2} - 3554 T^{3} + 49735 T^{4} - 449836 T^{5} + 115690 p T^{6} - 37513844 T^{7} + 327083851 T^{8} - 2326331990 T^{9} + 17697320220 T^{10} - 115153987482 T^{11} + 790833981402 T^{12} - 115153987482 p T^{13} + 17697320220 p^{2} T^{14} - 2326331990 p^{3} T^{15} + 327083851 p^{4} T^{16} - 37513844 p^{5} T^{17} + 115690 p^{7} T^{18} - 449836 p^{7} T^{19} + 49735 p^{8} T^{20} - 3554 p^{9} T^{21} + 330 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 15 T + 6 p T^{2} - 1904 T^{3} + 16807 T^{4} - 43956 T^{5} + 334214 T^{6} + 930811 T^{7} + 18567651 T^{8} - 73256050 T^{9} + 1947741960 T^{10} - 8094843132 T^{11} + 100281619882 T^{12} - 8094843132 p T^{13} + 1947741960 p^{2} T^{14} - 73256050 p^{3} T^{15} + 18567651 p^{4} T^{16} + 930811 p^{5} T^{17} + 334214 p^{6} T^{18} - 43956 p^{7} T^{19} + 16807 p^{8} T^{20} - 1904 p^{9} T^{21} + 6 p^{11} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 17 T + 416 T^{2} - 5222 T^{3} + 75152 T^{4} - 755072 T^{5} + 8230488 T^{6} - 69868476 T^{7} + 639405486 T^{8} - 4787353750 T^{9} + 38964096514 T^{10} - 265590831261 T^{11} + 1982556678638 T^{12} - 265590831261 p T^{13} + 38964096514 p^{2} T^{14} - 4787353750 p^{3} T^{15} + 639405486 p^{4} T^{16} - 69868476 p^{5} T^{17} + 8230488 p^{6} T^{18} - 755072 p^{7} T^{19} + 75152 p^{8} T^{20} - 5222 p^{9} T^{21} + 416 p^{10} T^{22} - 17 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 7 T + 90 T^{2} + 488 T^{3} + 12865 T^{4} + 79310 T^{5} + 853506 T^{6} + 3921193 T^{7} + 65117699 T^{8} + 321866540 T^{9} + 3600066580 T^{10} + 15238294818 T^{11} + 207282814582 T^{12} + 15238294818 p T^{13} + 3600066580 p^{2} T^{14} + 321866540 p^{3} T^{15} + 65117699 p^{4} T^{16} + 3921193 p^{5} T^{17} + 853506 p^{6} T^{18} + 79310 p^{7} T^{19} + 12865 p^{8} T^{20} + 488 p^{9} T^{21} + 90 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 32 T + 710 T^{2} - 11554 T^{3} + 163099 T^{4} - 1999788 T^{5} + 22569414 T^{6} - 232762054 T^{7} + 2268422075 T^{8} - 20712796662 T^{9} + 180762854212 T^{10} - 1489434396946 T^{11} + 11760399511698 T^{12} - 1489434396946 p T^{13} + 180762854212 p^{2} T^{14} - 20712796662 p^{3} T^{15} + 2268422075 p^{4} T^{16} - 232762054 p^{5} T^{17} + 22569414 p^{6} T^{18} - 1999788 p^{7} T^{19} + 163099 p^{8} T^{20} - 11554 p^{9} T^{21} + 710 p^{10} T^{22} - 32 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 8 T + 593 T^{2} + 4174 T^{3} + 164810 T^{4} + 1024441 T^{5} + 28619615 T^{6} + 157483913 T^{7} + 3485187899 T^{8} + 16996065526 T^{9} + 315625078990 T^{10} + 1362184283202 T^{11} + 21896493361504 T^{12} + 1362184283202 p T^{13} + 315625078990 p^{2} T^{14} + 16996065526 p^{3} T^{15} + 3485187899 p^{4} T^{16} + 157483913 p^{5} T^{17} + 28619615 p^{6} T^{18} + 1024441 p^{7} T^{19} + 164810 p^{8} T^{20} + 4174 p^{9} T^{21} + 593 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 39 T + 1027 T^{2} - 19953 T^{3} + 327827 T^{4} - 4668260 T^{5} + 60334624 T^{6} - 711808264 T^{7} + 7810781633 T^{8} - 79587200761 T^{9} + 760914720237 T^{10} - 6811012689499 T^{11} + 57514505185654 T^{12} - 6811012689499 p T^{13} + 760914720237 p^{2} T^{14} - 79587200761 p^{3} T^{15} + 7810781633 p^{4} T^{16} - 711808264 p^{5} T^{17} + 60334624 p^{6} T^{18} - 4668260 p^{7} T^{19} + 327827 p^{8} T^{20} - 19953 p^{9} T^{21} + 1027 p^{10} T^{22} - 39 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 35 T + 965 T^{2} - 18827 T^{3} + 4508 p T^{4} - 4564809 T^{5} + 59233681 T^{6} - 683322497 T^{7} + 7374100007 T^{8} - 73103273734 T^{9} + 693289648154 T^{10} - 6177140226134 T^{11} + 53453933493352 T^{12} - 6177140226134 p T^{13} + 693289648154 p^{2} T^{14} - 73103273734 p^{3} T^{15} + 7374100007 p^{4} T^{16} - 683322497 p^{5} T^{17} + 59233681 p^{6} T^{18} - 4564809 p^{7} T^{19} + 4508 p^{9} T^{20} - 18827 p^{9} T^{21} + 965 p^{10} T^{22} - 35 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 12 T + 369 T^{2} + 3938 T^{3} + 76243 T^{4} + 768288 T^{5} + 11261961 T^{6} + 104835486 T^{7} + 1302550371 T^{8} + 11200420910 T^{9} + 123899449814 T^{10} + 976028791990 T^{11} + 9844713826450 T^{12} + 976028791990 p T^{13} + 123899449814 p^{2} T^{14} + 11200420910 p^{3} T^{15} + 1302550371 p^{4} T^{16} + 104835486 p^{5} T^{17} + 11261961 p^{6} T^{18} + 768288 p^{7} T^{19} + 76243 p^{8} T^{20} + 3938 p^{9} T^{21} + 369 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 36 T + 1156 T^{2} - 26026 T^{3} + 521709 T^{4} - 8832620 T^{5} + 135692808 T^{6} - 1863508050 T^{7} + 23540265499 T^{8} - 272195166550 T^{9} + 2918160542324 T^{10} - 28938160763594 T^{11} + 267029484011502 T^{12} - 28938160763594 p T^{13} + 2918160542324 p^{2} T^{14} - 272195166550 p^{3} T^{15} + 23540265499 p^{4} T^{16} - 1863508050 p^{5} T^{17} + 135692808 p^{6} T^{18} - 8832620 p^{7} T^{19} + 521709 p^{8} T^{20} - 26026 p^{9} T^{21} + 1156 p^{10} T^{22} - 36 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 25 T + 783 T^{2} - 14775 T^{3} + 287637 T^{4} - 4408358 T^{5} + 66832046 T^{6} - 866074036 T^{7} + 10989644649 T^{8} - 123351930903 T^{9} + 1350839330659 T^{10} - 13295933048579 T^{11} + 127458076112610 T^{12} - 13295933048579 p T^{13} + 1350839330659 p^{2} T^{14} - 123351930903 p^{3} T^{15} + 10989644649 p^{4} T^{16} - 866074036 p^{5} T^{17} + 66832046 p^{6} T^{18} - 4408358 p^{7} T^{19} + 287637 p^{8} T^{20} - 14775 p^{9} T^{21} + 783 p^{10} T^{22} - 25 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 23 T + 825 T^{2} - 177 p T^{3} + 330433 T^{4} - 5305955 T^{5} + 84153798 T^{6} - 1153158478 T^{7} + 15132045743 T^{8} - 179133937149 T^{9} + 2021024761073 T^{10} - 20856731697130 T^{11} + 205426750778462 T^{12} - 20856731697130 p T^{13} + 2021024761073 p^{2} T^{14} - 179133937149 p^{3} T^{15} + 15132045743 p^{4} T^{16} - 1153158478 p^{5} T^{17} + 84153798 p^{6} T^{18} - 5305955 p^{7} T^{19} + 330433 p^{8} T^{20} - 177 p^{10} T^{21} + 825 p^{10} T^{22} - 23 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 2 T + 599 T^{2} + 1134 T^{3} + 186500 T^{4} + 298077 T^{5} + 39910011 T^{6} + 53566567 T^{7} + 6513962669 T^{8} + 7655493132 T^{9} + 851815353522 T^{10} + 900196244348 T^{11} + 91127580291916 T^{12} + 900196244348 p T^{13} + 851815353522 p^{2} T^{14} + 7655493132 p^{3} T^{15} + 6513962669 p^{4} T^{16} + 53566567 p^{5} T^{17} + 39910011 p^{6} T^{18} + 298077 p^{7} T^{19} + 186500 p^{8} T^{20} + 1134 p^{9} T^{21} + 599 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.10878963575651889100080116708, −2.87776047521485678437424041951, −2.78727783506958649067300501946, −2.59312066799484741894494479899, −2.54869721513187973697415170072, −2.49133921643382342275439815806, −2.39004092848671382524036031416, −2.35153618009159501350758530403, −2.32793836256792444118385325388, −2.26067525107499966709948261831, −2.14342356436274328223028457879, −2.11262817308438600543630058062, −2.06340617116138616674702651801, −1.67798551894031443946600596847, −1.52816439262473537684503051667, −1.48751063948637277567447977154, −1.46095036590766252378723291797, −1.23341106195958649786124806048, −1.15582395808914764195834061500, −1.05866052080518607260243880067, −0.936981695654590475031906063505, −0.895928643721691959666107822279, −0.62905087182049633332248604445, −0.49181182522703575664438345486, −0.35277196741851137542048016380, 0.35277196741851137542048016380, 0.49181182522703575664438345486, 0.62905087182049633332248604445, 0.895928643721691959666107822279, 0.936981695654590475031906063505, 1.05866052080518607260243880067, 1.15582395808914764195834061500, 1.23341106195958649786124806048, 1.46095036590766252378723291797, 1.48751063948637277567447977154, 1.52816439262473537684503051667, 1.67798551894031443946600596847, 2.06340617116138616674702651801, 2.11262817308438600543630058062, 2.14342356436274328223028457879, 2.26067525107499966709948261831, 2.32793836256792444118385325388, 2.35153618009159501350758530403, 2.39004092848671382524036031416, 2.49133921643382342275439815806, 2.54869721513187973697415170072, 2.59312066799484741894494479899, 2.78727783506958649067300501946, 2.87776047521485678437424041951, 3.10878963575651889100080116708
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.