Dirichlet series
| L(s) = 1 | + 60·4-s − 486·9-s + 580·16-s − 520·19-s + 4.24e3·29-s + 4.52e3·31-s − 2.91e4·36-s + 1.62e4·41-s − 1.44e4·49-s − 5.13e4·59-s + 8.23e4·61-s − 4.01e4·64-s + 3.51e4·71-s − 3.12e4·76-s + 1.94e4·79-s + 1.37e5·81-s + 6.60e5·89-s − 1.44e5·101-s + 3.12e5·109-s + 2.54e5·116-s − 7.81e5·121-s + 2.71e5·124-s + 127-s + 131-s + 137-s + 139-s − 2.81e5·144-s + ⋯ |
| L(s) = 1 | + 15/8·4-s − 2·9-s + 0.566·16-s − 0.330·19-s + 0.937·29-s + 0.845·31-s − 3.75·36-s + 1.50·41-s − 6/7·49-s − 1.91·59-s + 2.83·61-s − 1.22·64-s + 0.827·71-s − 0.619·76-s + 0.351·79-s + 7/3·81-s + 8.84·89-s − 1.40·101-s + 2.51·109-s + 1.75·116-s − 4.85·121-s + 1.58·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s − 1.13·144-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{12} \cdot 5^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{12} \cdot 5^{24} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.27010\times 10^{23}\) |
| Root analytic conductor: | \(9.17613\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{12} \cdot 5^{24} \cdot 7^{12} ,\ ( \ : [5/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(1.531949499\) |
| \(L(\frac12)\) | \(\approx\) | \(1.531949499\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 + p^{4} T^{2} )^{6} \) |
| 5 | \( 1 \) | |
| 7 | \( ( 1 + p^{4} T^{2} )^{6} \) | |
| good | 2 | \( 1 - 15 p^{2} T^{2} + 755 p^{2} T^{4} - 106247 T^{6} + 96295 p^{5} T^{8} - 662435 p^{7} T^{10} + 42398921 p^{6} T^{12} - 662435 p^{17} T^{14} + 96295 p^{25} T^{16} - 106247 p^{30} T^{18} + 755 p^{42} T^{20} - 15 p^{52} T^{22} + p^{60} T^{24} \) |
| 11 | \( ( 1 + 390780 T^{2} + 30290676 T^{3} + 86305104280 T^{4} - 87164640360 T^{5} + 16728347551945346 T^{6} - 87164640360 p^{5} T^{7} + 86305104280 p^{10} T^{8} + 30290676 p^{15} T^{9} + 390780 p^{20} T^{10} + p^{30} T^{12} )^{2} \) | |
| 13 | \( 1 - 1590330 T^{2} + 1035030464615 T^{4} - 369096368555162178 T^{6} + \)\(12\!\cdots\!55\)\( T^{8} - \)\(69\!\cdots\!40\)\( T^{10} + \)\(32\!\cdots\!94\)\( T^{12} - \)\(69\!\cdots\!40\)\( p^{10} T^{14} + \)\(12\!\cdots\!55\)\( p^{20} T^{16} - 369096368555162178 p^{30} T^{18} + 1035030464615 p^{40} T^{20} - 1590330 p^{50} T^{22} + p^{60} T^{24} \) | |
| 17 | \( 1 - 433706 p T^{2} + 28438967055063 T^{4} - 77169059245526941298 T^{6} + \)\(16\!\cdots\!35\)\( T^{8} - \)\(29\!\cdots\!56\)\( T^{10} + \)\(45\!\cdots\!02\)\( T^{12} - \)\(29\!\cdots\!56\)\( p^{10} T^{14} + \)\(16\!\cdots\!35\)\( p^{20} T^{16} - 77169059245526941298 p^{30} T^{18} + 28438967055063 p^{40} T^{20} - 433706 p^{51} T^{22} + p^{60} T^{24} \) | |
| 19 | \( ( 1 + 260 T + 9360238 T^{2} + 2911370364 T^{3} + 42567448684743 T^{4} + 16013427452531176 T^{5} + \)\(12\!\cdots\!20\)\( T^{6} + 16013427452531176 p^{5} T^{7} + 42567448684743 p^{10} T^{8} + 2911370364 p^{15} T^{9} + 9360238 p^{20} T^{10} + 260 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 23 | \( 1 - 26541214 T^{2} + 337037088584925 T^{4} - \)\(25\!\cdots\!74\)\( T^{6} + \)\(14\!\cdots\!62\)\( T^{8} - \)\(72\!\cdots\!78\)\( T^{10} + \)\(42\!\cdots\!49\)\( T^{12} - \)\(72\!\cdots\!78\)\( p^{10} T^{14} + \)\(14\!\cdots\!62\)\( p^{20} T^{16} - \)\(25\!\cdots\!74\)\( p^{30} T^{18} + 337037088584925 p^{40} T^{20} - 26541214 p^{50} T^{22} + p^{60} T^{24} \) | |
| 29 | \( ( 1 - 2122 T + 61927729 T^{2} - 60319384550 T^{3} + 1749059932447290 T^{4} - 286130535022567682 T^{5} + \)\(36\!\cdots\!69\)\( T^{6} - 286130535022567682 p^{5} T^{7} + 1749059932447290 p^{10} T^{8} - 60319384550 p^{15} T^{9} + 61927729 p^{20} T^{10} - 2122 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 31 | \( ( 1 - 2262 T + 112419247 T^{2} - 417926877690 T^{3} + 5856474088672179 T^{4} - 25664805201806807460 T^{5} + \)\(19\!\cdots\!66\)\( T^{6} - 25664805201806807460 p^{5} T^{7} + 5856474088672179 p^{10} T^{8} - 417926877690 p^{15} T^{9} + 112419247 p^{20} T^{10} - 2262 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 37 | \( 1 - 306961072 T^{2} + 41981638625021968 T^{4} - \)\(41\!\cdots\!88\)\( T^{6} + \)\(38\!\cdots\!80\)\( T^{8} - \)\(31\!\cdots\!36\)\( T^{10} + \)\(16\!\cdots\!58\)\( p^{2} T^{12} - \)\(31\!\cdots\!36\)\( p^{10} T^{14} + \)\(38\!\cdots\!80\)\( p^{20} T^{16} - \)\(41\!\cdots\!88\)\( p^{30} T^{18} + 41981638625021968 p^{40} T^{20} - 306961072 p^{50} T^{22} + p^{60} T^{24} \) | |
| 41 | \( ( 1 - 8124 T + 356907937 T^{2} - 3577147730508 T^{3} + 80202267374272203 T^{4} - 16010478551912658648 p T^{5} + \)\(11\!\cdots\!18\)\( T^{6} - 16010478551912658648 p^{6} T^{7} + 80202267374272203 p^{10} T^{8} - 3577147730508 p^{15} T^{9} + 356907937 p^{20} T^{10} - 8124 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 43 | \( 1 - 259670822 T^{2} + 53928464977990165 T^{4} - \)\(71\!\cdots\!90\)\( T^{6} + \)\(12\!\cdots\!06\)\( T^{8} - \)\(22\!\cdots\!70\)\( T^{10} + \)\(38\!\cdots\!81\)\( T^{12} - \)\(22\!\cdots\!70\)\( p^{10} T^{14} + \)\(12\!\cdots\!06\)\( p^{20} T^{16} - \)\(71\!\cdots\!90\)\( p^{30} T^{18} + 53928464977990165 p^{40} T^{20} - 259670822 p^{50} T^{22} + p^{60} T^{24} \) | |
| 47 | \( 1 - 891165452 T^{2} + 519273069379579650 T^{4} - \)\(21\!\cdots\!36\)\( T^{6} + \)\(72\!\cdots\!07\)\( T^{8} - \)\(20\!\cdots\!72\)\( T^{10} + \)\(49\!\cdots\!48\)\( T^{12} - \)\(20\!\cdots\!72\)\( p^{10} T^{14} + \)\(72\!\cdots\!07\)\( p^{20} T^{16} - \)\(21\!\cdots\!36\)\( p^{30} T^{18} + 519273069379579650 p^{40} T^{20} - 891165452 p^{50} T^{22} + p^{60} T^{24} \) | |
| 53 | \( 1 + 135999766 T^{2} + 17566596889434291 p T^{4} + \)\(11\!\cdots\!90\)\( T^{6} + \)\(37\!\cdots\!99\)\( T^{8} + \)\(40\!\cdots\!00\)\( T^{10} + \)\(86\!\cdots\!54\)\( T^{12} + \)\(40\!\cdots\!00\)\( p^{10} T^{14} + \)\(37\!\cdots\!99\)\( p^{20} T^{16} + \)\(11\!\cdots\!90\)\( p^{30} T^{18} + 17566596889434291 p^{41} T^{20} + 135999766 p^{50} T^{22} + p^{60} T^{24} \) | |
| 59 | \( ( 1 + 25658 T + 2943968003 T^{2} + 73124332719102 T^{3} + 4023386576112897215 T^{4} + \)\(94\!\cdots\!84\)\( T^{5} + \)\(34\!\cdots\!62\)\( T^{6} + \)\(94\!\cdots\!84\)\( p^{5} T^{7} + 4023386576112897215 p^{10} T^{8} + 73124332719102 p^{15} T^{9} + 2943968003 p^{20} T^{10} + 25658 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 61 | \( ( 1 - 41188 T + 1696241441 T^{2} - 31441722797780 T^{3} + 1943575237355487795 T^{4} - \)\(91\!\cdots\!28\)\( p T^{5} + \)\(24\!\cdots\!74\)\( T^{6} - \)\(91\!\cdots\!28\)\( p^{6} T^{7} + 1943575237355487795 p^{10} T^{8} - 31441722797780 p^{15} T^{9} + 1696241441 p^{20} T^{10} - 41188 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 67 | \( 1 - 5670366168 T^{2} + 16421099319694820904 T^{4} - \)\(34\!\cdots\!00\)\( T^{6} + \)\(58\!\cdots\!40\)\( T^{8} - \)\(87\!\cdots\!68\)\( T^{10} + \)\(12\!\cdots\!14\)\( T^{12} - \)\(87\!\cdots\!68\)\( p^{10} T^{14} + \)\(58\!\cdots\!40\)\( p^{20} T^{16} - \)\(34\!\cdots\!00\)\( p^{30} T^{18} + 16421099319694820904 p^{40} T^{20} - 5670366168 p^{50} T^{22} + p^{60} T^{24} \) | |
| 71 | \( ( 1 - 17580 T + 8359525656 T^{2} - 115915456938000 T^{3} + 32391716003299722640 T^{4} - \)\(36\!\cdots\!00\)\( T^{5} + \)\(74\!\cdots\!70\)\( T^{6} - \)\(36\!\cdots\!00\)\( p^{5} T^{7} + 32391716003299722640 p^{10} T^{8} - 115915456938000 p^{15} T^{9} + 8359525656 p^{20} T^{10} - 17580 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 73 | \( 1 - 5403360364 T^{2} + 22567198878994693858 T^{4} - \)\(66\!\cdots\!60\)\( T^{6} + \)\(17\!\cdots\!59\)\( T^{8} - \)\(40\!\cdots\!00\)\( T^{10} + \)\(85\!\cdots\!64\)\( T^{12} - \)\(40\!\cdots\!00\)\( p^{10} T^{14} + \)\(17\!\cdots\!59\)\( p^{20} T^{16} - \)\(66\!\cdots\!60\)\( p^{30} T^{18} + 22567198878994693858 p^{40} T^{20} - 5403360364 p^{50} T^{22} + p^{60} T^{24} \) | |
| 79 | \( ( 1 - 9736 T + 13248072184 T^{2} - 229894654482540 T^{3} + 81015683687577966420 T^{4} - \)\(16\!\cdots\!56\)\( T^{5} + \)\(30\!\cdots\!46\)\( T^{6} - \)\(16\!\cdots\!56\)\( p^{5} T^{7} + 81015683687577966420 p^{10} T^{8} - 229894654482540 p^{15} T^{9} + 13248072184 p^{20} T^{10} - 9736 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 83 | \( 1 - 22151116442 T^{2} + \)\(27\!\cdots\!79\)\( T^{4} - \)\(24\!\cdots\!50\)\( T^{6} + \)\(16\!\cdots\!15\)\( T^{8} - \)\(87\!\cdots\!32\)\( T^{10} + \)\(37\!\cdots\!34\)\( T^{12} - \)\(87\!\cdots\!32\)\( p^{10} T^{14} + \)\(16\!\cdots\!15\)\( p^{20} T^{16} - \)\(24\!\cdots\!50\)\( p^{30} T^{18} + \)\(27\!\cdots\!79\)\( p^{40} T^{20} - 22151116442 p^{50} T^{22} + p^{60} T^{24} \) | |
| 89 | \( ( 1 - 330360 T + 62134968718 T^{2} - 8497860201285384 T^{3} + \)\(93\!\cdots\!83\)\( T^{4} - \)\(86\!\cdots\!16\)\( T^{5} + \)\(69\!\cdots\!20\)\( T^{6} - \)\(86\!\cdots\!16\)\( p^{5} T^{7} + \)\(93\!\cdots\!83\)\( p^{10} T^{8} - 8497860201285384 p^{15} T^{9} + 62134968718 p^{20} T^{10} - 330360 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 97 | \( 1 - 47286820916 T^{2} + 12402209220988118914 p T^{4} - \)\(21\!\cdots\!20\)\( T^{6} + \)\(30\!\cdots\!59\)\( T^{8} - \)\(34\!\cdots\!60\)\( T^{10} + \)\(31\!\cdots\!64\)\( T^{12} - \)\(34\!\cdots\!60\)\( p^{10} T^{14} + \)\(30\!\cdots\!59\)\( p^{20} T^{16} - \)\(21\!\cdots\!20\)\( p^{30} T^{18} + 12402209220988118914 p^{41} T^{20} - 47286820916 p^{50} T^{22} + p^{60} T^{24} \) | |
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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
−2.79086312439363875614255000196, −2.49121191594385414627667453874, −2.39920281271969837659374616580, −2.37853909832134141389236456380, −2.33148844083656623959586195387, −2.15317202746954430855086723640, −2.05607382475955297268413901274, −2.05384119257252761984516064690, −2.05194729677335140142966513815, −1.85920704280949901949077764427, −1.75997125717012737791769436253, −1.74659510180977945415552404320, −1.52873716899968019658569405582, −1.20689984924819612232525805766, −1.05685852224991619005722305878, −0.982494362275771763208009032271, −0.969045397316392129789027464847, −0.934114940871519556586373697800, −0.924973530722190663357820411004, −0.77850027269007296612732416984, −0.51516296229740005135891815512, −0.44312141595758585985318560560, −0.23423928429574645888724322563, −0.098653564951761254446302196143, −0.085086990085508227332287752714, 0.085086990085508227332287752714, 0.098653564951761254446302196143, 0.23423928429574645888724322563, 0.44312141595758585985318560560, 0.51516296229740005135891815512, 0.77850027269007296612732416984, 0.924973530722190663357820411004, 0.934114940871519556586373697800, 0.969045397316392129789027464847, 0.982494362275771763208009032271, 1.05685852224991619005722305878, 1.20689984924819612232525805766, 1.52873716899968019658569405582, 1.74659510180977945415552404320, 1.75997125717012737791769436253, 1.85920704280949901949077764427, 2.05194729677335140142966513815, 2.05384119257252761984516064690, 2.05607382475955297268413901274, 2.15317202746954430855086723640, 2.33148844083656623959586195387, 2.37853909832134141389236456380, 2.39920281271969837659374616580, 2.49121191594385414627667453874, 2.79086312439363875614255000196
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.