Dirichlet series
| L(s) = 1 | + 18·5-s − 27·9-s − 6·17-s + 237·25-s − 114·37-s − 486·45-s + 90·49-s − 18·53-s + 318·61-s + 342·73-s + 318·81-s − 108·85-s + 150·89-s + 498·101-s − 318·109-s − 1.34e3·113-s − 363·121-s + 2.47e3·125-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 162·153-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 18/5·5-s − 3·9-s − 0.352·17-s + 9.47·25-s − 3.08·37-s − 10.7·45-s + 1.83·49-s − 0.339·53-s + 5.21·61-s + 4.68·73-s + 3.92·81-s − 1.27·85-s + 1.68·89-s + 4.93·101-s − 2.91·109-s − 11.8·113-s − 3·121-s + 19.7·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 1.05·153-s + 0.00636·157-s + 0.00613·163-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{60} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{60} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.67298\times 10^{9}\) |
| Root analytic conductor: | \(2.47053\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{60} \cdot 7^{12} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(6.698059337\) |
| \(L(\frac12)\) | \(\approx\) | \(6.698059337\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 90 T^{2} + 159 p^{2} T^{4} - 172 p^{4} T^{6} + 159 p^{6} T^{8} - 90 p^{8} T^{10} + p^{12} T^{12} \) | |
| good | 3 | \( 1 + p^{3} T^{2} + 137 p T^{4} + 2972 T^{6} - 47 p^{2} T^{8} - 107053 p T^{10} - 4031882 T^{12} - 107053 p^{5} T^{14} - 47 p^{10} T^{16} + 2972 p^{12} T^{18} + 137 p^{17} T^{20} + p^{23} T^{22} + p^{24} T^{24} \) |
| 5 | \( ( 1 - 9 T + 3 T^{2} + 28 p T^{3} + 69 p T^{4} - 3363 T^{5} + 5766 T^{6} - 3363 p^{2} T^{7} + 69 p^{5} T^{8} + 28 p^{7} T^{9} + 3 p^{8} T^{10} - 9 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 11 | \( 1 + 3 p^{2} T^{2} + 55851 T^{4} + 5892268 T^{6} + 719654457 T^{8} + 93270469209 T^{10} + 11000493019158 T^{12} + 93270469209 p^{4} T^{14} + 719654457 p^{8} T^{16} + 5892268 p^{12} T^{18} + 55851 p^{16} T^{20} + 3 p^{22} T^{22} + p^{24} T^{24} \) | |
| 13 | \( ( 1 + 339 T^{2} - 784 T^{3} + 339 p^{2} T^{4} + p^{6} T^{6} )^{4} \) | |
| 17 | \( ( 1 + 3 T - 261 T^{2} - 5460 T^{3} - 855 p T^{4} + 628833 T^{5} + 36283574 T^{6} + 628833 p^{2} T^{7} - 855 p^{5} T^{8} - 5460 p^{6} T^{9} - 261 p^{8} T^{10} + 3 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 19 | \( 1 + 411 T^{2} - 128661 T^{4} - 89218804 T^{6} + 6305474169 T^{8} + 6575128475913 T^{10} + 1202874182390166 T^{12} + 6575128475913 p^{4} T^{14} + 6305474169 p^{8} T^{16} - 89218804 p^{12} T^{18} - 128661 p^{16} T^{20} + 411 p^{20} T^{22} + p^{24} T^{24} \) | |
| 23 | \( 1 + 579 T^{2} + 2733 p T^{4} - 35156516 T^{6} - 57866315751 T^{8} - 16658355952143 T^{10} + 6113539249932246 T^{12} - 16658355952143 p^{4} T^{14} - 57866315751 p^{8} T^{16} - 35156516 p^{12} T^{18} + 2733 p^{17} T^{20} + 579 p^{20} T^{22} + p^{24} T^{24} \) | |
| 29 | \( ( 1 + 1347 T^{2} - 5488 T^{3} + 1347 p^{2} T^{4} + p^{6} T^{6} )^{4} \) | |
| 31 | \( 1 + 5235 T^{2} + 15533691 T^{4} + 32305589068 T^{6} + 51858557764857 T^{8} + 66810392242083729 T^{10} + 70681395526523023542 T^{12} + 66810392242083729 p^{4} T^{14} + 51858557764857 p^{8} T^{16} + 32305589068 p^{12} T^{18} + 15533691 p^{16} T^{20} + 5235 p^{20} T^{22} + p^{24} T^{24} \) | |
| 37 | \( ( 1 + 57 T - 1269 T^{2} - 55332 T^{3} + 5310729 T^{4} + 108467211 T^{5} - 4432135786 T^{6} + 108467211 p^{2} T^{7} + 5310729 p^{4} T^{8} - 55332 p^{6} T^{9} - 1269 p^{8} T^{10} + 57 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 41 | \( ( 1 + 3195 T^{2} + 22736 T^{3} + 3195 p^{2} T^{4} + p^{6} T^{6} )^{4} \) | |
| 43 | \( ( 1 - 5718 T^{2} + 18746463 T^{4} - 40352107060 T^{6} + 18746463 p^{4} T^{8} - 5718 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 47 | \( 1 + 8307 T^{2} + 34424571 T^{4} + 100893075532 T^{6} + 253028343980217 T^{8} + 616841824418715921 T^{10} + \)\(14\!\cdots\!38\)\( T^{12} + 616841824418715921 p^{4} T^{14} + 253028343980217 p^{8} T^{16} + 100893075532 p^{12} T^{18} + 34424571 p^{16} T^{20} + 8307 p^{20} T^{22} + p^{24} T^{24} \) | |
| 53 | \( ( 1 + 9 T - 2997 T^{2} - 116484 T^{3} + 179145 T^{4} + 137286747 T^{5} + 22274291990 T^{6} + 137286747 p^{2} T^{7} + 179145 p^{4} T^{8} - 116484 p^{6} T^{9} - 2997 p^{8} T^{10} + 9 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 59 | \( 1 + 1707 T^{2} - 9128709 T^{4} - 112178055524 T^{6} - 114189887648679 T^{8} + 551687617569890985 T^{10} + \)\(57\!\cdots\!54\)\( T^{12} + 551687617569890985 p^{4} T^{14} - 114189887648679 p^{8} T^{16} - 112178055524 p^{12} T^{18} - 9128709 p^{16} T^{20} + 1707 p^{20} T^{22} + p^{24} T^{24} \) | |
| 61 | \( ( 1 - 159 T + 7899 T^{2} - 401828 T^{3} + 49683657 T^{4} - 2161110333 T^{5} + 23418423222 T^{6} - 2161110333 p^{2} T^{7} + 49683657 p^{4} T^{8} - 401828 p^{6} T^{9} + 7899 p^{8} T^{10} - 159 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 67 | \( 1 + 23547 T^{2} + 312061755 T^{4} + 2891423739964 T^{6} + 20714504811186393 T^{8} + \)\(12\!\cdots\!29\)\( T^{10} + \)\(59\!\cdots\!86\)\( T^{12} + \)\(12\!\cdots\!29\)\( p^{4} T^{14} + 20714504811186393 p^{8} T^{16} + 2891423739964 p^{12} T^{18} + 312061755 p^{16} T^{20} + 23547 p^{20} T^{22} + p^{24} T^{24} \) | |
| 71 | \( ( 1 - 10086 T^{2} + 56255151 T^{4} - 272956768468 T^{6} + 56255151 p^{4} T^{8} - 10086 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 73 | \( ( 1 - 171 T + 4275 T^{2} - 355908 T^{3} + 155672073 T^{4} - 112391289 p T^{5} - 20860196506 T^{6} - 112391289 p^{3} T^{7} + 155672073 p^{4} T^{8} - 355908 p^{6} T^{9} + 4275 p^{8} T^{10} - 171 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 79 | \( 1 + 16371 T^{2} + 121567851 T^{4} + 486780511900 T^{6} + 101914373627865 T^{8} - 22023984316104222399 T^{10} - \)\(21\!\cdots\!94\)\( T^{12} - 22023984316104222399 p^{4} T^{14} + 101914373627865 p^{8} T^{16} + 486780511900 p^{12} T^{18} + 121567851 p^{16} T^{20} + 16371 p^{20} T^{22} + p^{24} T^{24} \) | |
| 83 | \( ( 1 - 29238 T^{2} + 4861989 p T^{4} - 3424551164404 T^{6} + 4861989 p^{5} T^{8} - 29238 p^{8} T^{10} + p^{12} T^{12} )^{2} \) | |
| 89 | \( ( 1 - 75 T - 7725 T^{2} + 660828 T^{3} + 10149225 T^{4} + 693975 p T^{5} - 80037126682 T^{6} + 693975 p^{3} T^{7} + 10149225 p^{4} T^{8} + 660828 p^{6} T^{9} - 7725 p^{8} T^{10} - 75 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 97 | \( ( 1 + 28059 T^{2} - 784 T^{3} + 28059 p^{2} T^{4} + p^{6} T^{6} )^{4} \) | |
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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
−3.88788568199015792365629860147, −3.76663637316202300538660139505, −3.76364633857320158093375293687, −3.69809514791302610944476711405, −3.40126733596866771551526150816, −3.38785475657359894479319269093, −3.30463960351991159074393286879, −2.92969716382321068127188953426, −2.83049238785455048134068137861, −2.71942937061422165779340503494, −2.64968218065577944596736951547, −2.61065372373380724242104585947, −2.39400933499699871607615840961, −2.34162535743241463505707340252, −2.33403517947043855015322225223, −2.06629997327496561949769314949, −1.95961172870318299764261903827, −1.77081372286763083070868000157, −1.44736091188005246314943679390, −1.34890551124649773758740997079, −1.12465570432115330499517115919, −0.936267378308602813430158530324, −0.921954187591722276703480926892, −0.35661053400609934463062093197, −0.22411971436784255627757975881, 0.22411971436784255627757975881, 0.35661053400609934463062093197, 0.921954187591722276703480926892, 0.936267378308602813430158530324, 1.12465570432115330499517115919, 1.34890551124649773758740997079, 1.44736091188005246314943679390, 1.77081372286763083070868000157, 1.95961172870318299764261903827, 2.06629997327496561949769314949, 2.33403517947043855015322225223, 2.34162535743241463505707340252, 2.39400933499699871607615840961, 2.61065372373380724242104585947, 2.64968218065577944596736951547, 2.71942937061422165779340503494, 2.83049238785455048134068137861, 2.92969716382321068127188953426, 3.30463960351991159074393286879, 3.38785475657359894479319269093, 3.40126733596866771551526150816, 3.69809514791302610944476711405, 3.76364633857320158093375293687, 3.76663637316202300538660139505, 3.88788568199015792365629860147
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.