Dirichlet series
| L(s) = 1 | + 6·3-s − 12·7-s + 15·9-s − 6·11-s + 6·13-s − 18·19-s − 72·21-s − 9·25-s + 16·27-s + 18·29-s − 36·33-s + 30·37-s + 36·39-s + 24·41-s + 6·47-s + 78·49-s − 12·53-s − 108·57-s − 36·61-s − 180·63-s + 64-s − 30·67-s + 12·71-s − 54·75-s + 72·77-s + 6·79-s + 3·81-s + ⋯ |
| L(s) = 1 | + 3.46·3-s − 4.53·7-s + 5·9-s − 1.80·11-s + 1.66·13-s − 4.12·19-s − 15.7·21-s − 9/5·25-s + 3.07·27-s + 3.34·29-s − 6.26·33-s + 4.93·37-s + 5.76·39-s + 3.74·41-s + 0.875·47-s + 78/7·49-s − 1.64·53-s − 14.3·57-s − 4.60·61-s − 22.6·63-s + 1/8·64-s − 3.66·67-s + 1.42·71-s − 6.23·75-s + 8.20·77-s + 0.675·79-s + 1/3·81-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 37^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.00181178\) |
| Root analytic conductor: | \(0.768695\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 37^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.6154193153\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6154193153\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - T^{6} + T^{12} \) |
| 37 | \( 1 - 30 T + 417 T^{2} - 3838 T^{3} + 27450 T^{4} - 164430 T^{5} + 953241 T^{6} - 164430 p T^{7} + 27450 p^{2} T^{8} - 3838 p^{3} T^{9} + 417 p^{4} T^{10} - 30 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 3 | \( ( 1 - p T + 2 p T^{2} - 8 T^{3} + p T^{4} + 7 p T^{5} - 53 T^{6} + 7 p^{2} T^{7} + p^{3} T^{8} - 8 p^{3} T^{9} + 2 p^{5} T^{10} - p^{6} T^{11} + p^{6} T^{12} )^{2} \) |
| 5 | \( 1 + 9 T^{2} + 18 T^{3} + 36 T^{4} + 108 T^{5} + 286 T^{6} + 162 T^{7} + 351 T^{8} + 252 p T^{9} - 5589 T^{10} - 1242 p T^{11} - 22119 T^{12} - 1242 p^{2} T^{13} - 5589 p^{2} T^{14} + 252 p^{4} T^{15} + 351 p^{4} T^{16} + 162 p^{5} T^{17} + 286 p^{6} T^{18} + 108 p^{7} T^{19} + 36 p^{8} T^{20} + 18 p^{9} T^{21} + 9 p^{10} T^{22} + p^{12} T^{24} \) | |
| 7 | \( 1 + 12 T + 66 T^{2} + 218 T^{3} + 66 p T^{4} + 12 p^{2} T^{5} + 289 T^{6} - 648 T^{7} - 4131 T^{8} - 22354 T^{9} - 327 p^{3} T^{10} - 455736 T^{11} - 1395491 T^{12} - 455736 p T^{13} - 327 p^{5} T^{14} - 22354 p^{3} T^{15} - 4131 p^{4} T^{16} - 648 p^{5} T^{17} + 289 p^{6} T^{18} + 12 p^{9} T^{19} + 66 p^{9} T^{20} + 218 p^{9} T^{21} + 66 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 11 | \( 1 + 6 T - 3 T^{2} - 126 T^{3} - 384 T^{4} + 78 T^{5} + 3947 T^{6} + 14724 T^{7} + 20151 T^{8} - 35478 T^{9} - 185580 T^{10} - 507132 T^{11} - 1637679 T^{12} - 507132 p T^{13} - 185580 p^{2} T^{14} - 35478 p^{3} T^{15} + 20151 p^{4} T^{16} + 14724 p^{5} T^{17} + 3947 p^{6} T^{18} + 78 p^{7} T^{19} - 384 p^{8} T^{20} - 126 p^{9} T^{21} - 3 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 6 T + 30 T^{2} + 60 T^{3} - 912 T^{4} + 5154 T^{5} - 8551 T^{6} - 47682 T^{7} + 30951 p T^{8} - 1393602 T^{9} + 47223 p T^{10} + 15564222 T^{11} - 89154855 T^{12} + 15564222 p T^{13} + 47223 p^{3} T^{14} - 1393602 p^{3} T^{15} + 30951 p^{5} T^{16} - 47682 p^{5} T^{17} - 8551 p^{6} T^{18} + 5154 p^{7} T^{19} - 912 p^{8} T^{20} + 60 p^{9} T^{21} + 30 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 36 T^{2} - 72 T^{3} + 576 T^{4} - 432 T^{5} + 7810 T^{6} + 17496 T^{7} - 83268 T^{8} + 7632 p T^{9} - 3484188 T^{10} + 6754320 T^{11} - 47006925 T^{12} + 6754320 p T^{13} - 3484188 p^{2} T^{14} + 7632 p^{4} T^{15} - 83268 p^{4} T^{16} + 17496 p^{5} T^{17} + 7810 p^{6} T^{18} - 432 p^{7} T^{19} + 576 p^{8} T^{20} - 72 p^{9} T^{21} + 36 p^{10} T^{22} + p^{12} T^{24} \) | |
| 19 | \( 1 + 18 T + 135 T^{2} + 36 p T^{3} + 3987 T^{4} + 22842 T^{5} + 86699 T^{6} + 237294 T^{7} + 682254 T^{8} + 221364 T^{9} - 17657496 T^{10} - 114009210 T^{11} - 491591103 T^{12} - 114009210 p T^{13} - 17657496 p^{2} T^{14} + 221364 p^{3} T^{15} + 682254 p^{4} T^{16} + 237294 p^{5} T^{17} + 86699 p^{6} T^{18} + 22842 p^{7} T^{19} + 3987 p^{8} T^{20} + 36 p^{10} T^{21} + 135 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 + 75 T^{2} + 2820 T^{4} - 486 T^{5} + 2843 p T^{6} - 88938 T^{7} + 1000341 T^{8} - 5307120 T^{9} + 9514314 T^{10} - 187937658 T^{11} + 84111261 T^{12} - 187937658 p T^{13} + 9514314 p^{2} T^{14} - 5307120 p^{3} T^{15} + 1000341 p^{4} T^{16} - 88938 p^{5} T^{17} + 2843 p^{7} T^{18} - 486 p^{7} T^{19} + 2820 p^{8} T^{20} + 75 p^{10} T^{22} + p^{12} T^{24} \) | |
| 29 | \( 1 - 18 T + 300 T^{2} - 3456 T^{3} + 37074 T^{4} - 328356 T^{5} + 2747362 T^{6} - 20241414 T^{7} + 142598466 T^{8} - 913009860 T^{9} + 5656241286 T^{10} - 32404114530 T^{11} + 180882283299 T^{12} - 32404114530 p T^{13} + 5656241286 p^{2} T^{14} - 913009860 p^{3} T^{15} + 142598466 p^{4} T^{16} - 20241414 p^{5} T^{17} + 2747362 p^{6} T^{18} - 328356 p^{7} T^{19} + 37074 p^{8} T^{20} - 3456 p^{9} T^{21} + 300 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 162 T^{2} + 15255 T^{4} - 1015396 T^{6} + 52463889 T^{8} - 2178592578 T^{10} + 74378126991 T^{12} - 2178592578 p^{2} T^{14} + 52463889 p^{4} T^{16} - 1015396 p^{6} T^{18} + 15255 p^{8} T^{20} - 162 p^{10} T^{22} + p^{12} T^{24} \) | |
| 41 | \( 1 - 24 T + 216 T^{2} - 828 T^{3} + 720 T^{4} - 13956 T^{5} + 391898 T^{6} - 3438108 T^{7} + 10896192 T^{8} + 13992048 T^{9} + 77900472 T^{10} - 5170021092 T^{11} + 50104727283 T^{12} - 5170021092 p T^{13} + 77900472 p^{2} T^{14} + 13992048 p^{3} T^{15} + 10896192 p^{4} T^{16} - 3438108 p^{5} T^{17} + 391898 p^{6} T^{18} - 13956 p^{7} T^{19} + 720 p^{8} T^{20} - 828 p^{9} T^{21} + 216 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 360 T^{2} + 63864 T^{4} - 7338454 T^{6} + 606798000 T^{8} - 38000588400 T^{10} + 1847557538571 T^{12} - 38000588400 p^{2} T^{14} + 606798000 p^{4} T^{16} - 7338454 p^{6} T^{18} + 63864 p^{8} T^{20} - 360 p^{10} T^{22} + p^{12} T^{24} \) | |
| 47 | \( 1 - 6 T - 129 T^{2} + 630 T^{3} + 7275 T^{4} - 13200 T^{5} - 10496 p T^{6} - 177264 T^{7} + 38861325 T^{8} - 36485370 T^{9} - 1800081855 T^{10} + 37175094 p T^{11} + 66999936462 T^{12} + 37175094 p^{2} T^{13} - 1800081855 p^{2} T^{14} - 36485370 p^{3} T^{15} + 38861325 p^{4} T^{16} - 177264 p^{5} T^{17} - 10496 p^{7} T^{18} - 13200 p^{7} T^{19} + 7275 p^{8} T^{20} + 630 p^{9} T^{21} - 129 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 12 T + 162 T^{2} + 2070 T^{3} + 17406 T^{4} + 184260 T^{5} + 1542269 T^{6} + 11580300 T^{7} + 102370437 T^{8} + 674033346 T^{9} + 5148381051 T^{10} + 39964816152 T^{11} + 251484462021 T^{12} + 39964816152 p T^{13} + 5148381051 p^{2} T^{14} + 674033346 p^{3} T^{15} + 102370437 p^{4} T^{16} + 11580300 p^{5} T^{17} + 1542269 p^{6} T^{18} + 184260 p^{7} T^{19} + 17406 p^{8} T^{20} + 2070 p^{9} T^{21} + 162 p^{10} T^{22} + 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 342 T^{3} + 1674 T^{4} - 49014 T^{5} - 53525 T^{6} + 79740 T^{7} - 16674903 T^{8} - 113688882 T^{9} + 1369196325 T^{10} + 3553369002 T^{11} - 10101568623 T^{12} + 3553369002 p T^{13} + 1369196325 p^{2} T^{14} - 113688882 p^{3} T^{15} - 16674903 p^{4} T^{16} + 79740 p^{5} T^{17} - 53525 p^{6} T^{18} - 49014 p^{7} T^{19} + 1674 p^{8} T^{20} + 342 p^{9} T^{21} + p^{12} T^{24} \) | |
| 61 | \( 1 + 36 T + 756 T^{2} + 12564 T^{3} + 176328 T^{4} + 2189844 T^{5} + 24855698 T^{6} + 261642168 T^{7} + 2584948968 T^{8} + 24136234056 T^{9} + 213689013264 T^{10} + 1797416529192 T^{11} + 14385250976619 T^{12} + 1797416529192 p T^{13} + 213689013264 p^{2} T^{14} + 24136234056 p^{3} T^{15} + 2584948968 p^{4} T^{16} + 261642168 p^{5} T^{17} + 24855698 p^{6} T^{18} + 2189844 p^{7} T^{19} + 176328 p^{8} T^{20} + 12564 p^{9} T^{21} + 756 p^{10} T^{22} + 36 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 30 T + 669 T^{2} + 10784 T^{3} + 145236 T^{4} + 1633170 T^{5} + 15989248 T^{6} + 136598616 T^{7} + 1029520098 T^{8} + 6927400208 T^{9} + 42351918369 T^{10} + 258461868804 T^{11} + 1824238562935 T^{12} + 258461868804 p T^{13} + 42351918369 p^{2} T^{14} + 6927400208 p^{3} T^{15} + 1029520098 p^{4} T^{16} + 136598616 p^{5} T^{17} + 15989248 p^{6} T^{18} + 1633170 p^{7} T^{19} + 145236 p^{8} T^{20} + 10784 p^{9} T^{21} + 669 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 12 T + 216 T^{2} - 2232 T^{3} + 360 p T^{4} - 171552 T^{5} + 1359938 T^{6} - 7875324 T^{7} + 67831200 T^{8} - 510242112 T^{9} + 8171477388 T^{10} - 73279337940 T^{11} + 777801762579 T^{12} - 73279337940 p T^{13} + 8171477388 p^{2} T^{14} - 510242112 p^{3} T^{15} + 67831200 p^{4} T^{16} - 7875324 p^{5} T^{17} + 1359938 p^{6} T^{18} - 171552 p^{7} T^{19} + 360 p^{9} T^{20} - 2232 p^{9} T^{21} + 216 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( ( 1 + 72 T^{2} + 322 T^{3} + 1368 T^{4} - 44892 T^{5} + 304995 T^{6} - 44892 p T^{7} + 1368 p^{2} T^{8} + 322 p^{3} T^{9} + 72 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 79 | \( 1 - 6 T + 12 T^{2} + 1248 T^{3} - 10650 T^{4} - 22728 T^{5} + 1578554 T^{6} - 11024046 T^{7} - 6224922 T^{8} + 1317987588 T^{9} - 6433288974 T^{10} - 46382041998 T^{11} + 1018948827267 T^{12} - 46382041998 p T^{13} - 6433288974 p^{2} T^{14} + 1317987588 p^{3} T^{15} - 6224922 p^{4} T^{16} - 11024046 p^{5} T^{17} + 1578554 p^{6} T^{18} - 22728 p^{7} T^{19} - 10650 p^{8} T^{20} + 1248 p^{9} T^{21} + 12 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 48 T + 1008 T^{2} + 11466 T^{3} + 60894 T^{4} - 91014 T^{5} - 1749697 T^{6} + 34462044 T^{7} + 8868375 p T^{8} + 6180808302 T^{9} + 30987329049 T^{10} + 170074677726 T^{11} + 1524826340769 T^{12} + 170074677726 p T^{13} + 30987329049 p^{2} T^{14} + 6180808302 p^{3} T^{15} + 8868375 p^{5} T^{16} + 34462044 p^{5} T^{17} - 1749697 p^{6} T^{18} - 91014 p^{7} T^{19} + 60894 p^{8} T^{20} + 11466 p^{9} T^{21} + 1008 p^{10} T^{22} + 48 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 18 T + 9 T^{2} - 1296 T^{3} + 16236 T^{4} + 284814 T^{5} - 1490888 T^{6} - 24667128 T^{7} + 210783978 T^{8} + 1425964500 T^{9} - 29432650887 T^{10} - 99080020404 T^{11} + 2142973122519 T^{12} - 99080020404 p T^{13} - 29432650887 p^{2} T^{14} + 1425964500 p^{3} T^{15} + 210783978 p^{4} T^{16} - 24667128 p^{5} T^{17} - 1490888 p^{6} T^{18} + 284814 p^{7} T^{19} + 16236 p^{8} T^{20} - 1296 p^{9} T^{21} + 9 p^{10} T^{22} + 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 36 T + 918 T^{2} - 17496 T^{3} + 280341 T^{4} - 3647664 T^{5} + 39168002 T^{6} - 317393748 T^{7} + 1369191978 T^{8} + 12571294932 T^{9} - 398485202130 T^{10} + 6007641045408 T^{11} - 66848579588931 T^{12} + 6007641045408 p T^{13} - 398485202130 p^{2} T^{14} + 12571294932 p^{3} T^{15} + 1369191978 p^{4} T^{16} - 317393748 p^{5} T^{17} + 39168002 p^{6} T^{18} - 3647664 p^{7} T^{19} + 280341 p^{8} T^{20} - 17496 p^{9} T^{21} + 918 p^{10} T^{22} - 36 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−5.76299559646189544573590098072, −5.65282805162515088236084507233, −5.18612393138735563812583515025, −4.96846762356247204586932777589, −4.79178088602117717488276735476, −4.65317876991440684369376646399, −4.57894080905580908287929945523, −4.49469237045594641533512962211, −4.21432052587269823651922698858, −4.16219124229754266687910926540, −4.06126131264194166663343268673, −4.05004939622091891098649431846, −3.57366065209058556957467984464, −3.56608600280887833588013475599, −3.32864567113728287034536323364, −3.23913226609077370373622393797, −3.15142266030834501031636738799, −2.85748680373103714221288870135, −2.76378722663148320127200442087, −2.74776678098280708047157717487, −2.54674771238042052571562338493, −2.30041708353806212635810130905, −2.21904528216493128998787109743, −2.13607921672580497508811577376, −1.23225079956428996471950766790, 1.23225079956428996471950766790, 2.13607921672580497508811577376, 2.21904528216493128998787109743, 2.30041708353806212635810130905, 2.54674771238042052571562338493, 2.74776678098280708047157717487, 2.76378722663148320127200442087, 2.85748680373103714221288870135, 3.15142266030834501031636738799, 3.23913226609077370373622393797, 3.32864567113728287034536323364, 3.56608600280887833588013475599, 3.57366065209058556957467984464, 4.05004939622091891098649431846, 4.06126131264194166663343268673, 4.16219124229754266687910926540, 4.21432052587269823651922698858, 4.49469237045594641533512962211, 4.57894080905580908287929945523, 4.65317876991440684369376646399, 4.79178088602117717488276735476, 4.96846762356247204586932777589, 5.18612393138735563812583515025, 5.65282805162515088236084507233, 5.76299559646189544573590098072
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.