Dirichlet series
| L(s) = 1 | − 3·2-s − 11·4-s − 24·5-s − 15·8-s + 72·10-s + 120·13-s − 3·16-s + 648·17-s + 264·20-s − 2.29e3·25-s − 360·26-s − 24·29-s + 729·32-s − 1.94e3·34-s − 424·37-s + 360·40-s − 7.32e3·41-s − 2.05e3·49-s + 6.87e3·50-s − 1.32e3·52-s + 7.08e3·53-s + 72·58-s + 9.75e3·61-s + 3.52e3·64-s − 2.88e3·65-s − 7.12e3·68-s + 1.37e4·73-s + ⋯ |
| L(s) = 1 | − 3/4·2-s − 0.687·4-s − 0.959·5-s − 0.234·8-s + 0.719·10-s + 0.710·13-s − 0.0117·16-s + 2.24·17-s + 0.659·20-s − 3.66·25-s − 0.532·26-s − 0.0285·29-s + 0.711·32-s − 1.68·34-s − 0.309·37-s + 9/40·40-s − 4.35·41-s − 6/7·49-s + 2.75·50-s − 0.488·52-s + 2.52·53-s + 0.0214·58-s + 2.62·61-s + 0.860·64-s − 0.681·65-s − 1.54·68-s + 2.58·73-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 3^{24} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.76200\times 10^{16}\) |
| Root analytic conductor: | \(5.10384\) |
| Motivic weight: | \(4\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 3^{24} \cdot 7^{12} ,\ ( \ : [2]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{5}{2})\) | \(\approx\) | \(0.09467100456\) |
| \(L(\frac12)\) | \(\approx\) | \(0.09467100456\) |
| \(L(3)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 + 3 T + 5 p^{2} T^{2} + 27 p^{2} T^{3} + 37 p^{4} T^{4} + 159 p^{4} T^{5} + 79 p^{7} T^{6} + 159 p^{8} T^{7} + 37 p^{12} T^{8} + 27 p^{14} T^{9} + 5 p^{18} T^{10} + 3 p^{20} T^{11} + p^{24} T^{12} \) |
| 3 | \( 1 \) | |
| 7 | \( ( 1 + p^{3} T^{2} )^{6} \) | |
| good | 5 | \( ( 1 + 12 T + 1362 T^{2} + 1404 p T^{3} + 682031 T^{4} - 13337016 T^{5} + 277954556 T^{6} - 13337016 p^{4} T^{7} + 682031 p^{8} T^{8} + 1404 p^{13} T^{9} + 1362 p^{16} T^{10} + 12 p^{20} T^{11} + p^{24} T^{12} )^{2} \) |
| 11 | \( 1 - 8164 p T^{2} + 4272882690 T^{4} - 139722687216412 T^{6} + 3477867716644402991 T^{8} - \)\(69\!\cdots\!16\)\( T^{10} + \)\(11\!\cdots\!36\)\( T^{12} - \)\(69\!\cdots\!16\)\( p^{8} T^{14} + 3477867716644402991 p^{16} T^{16} - 139722687216412 p^{24} T^{18} + 4272882690 p^{32} T^{20} - 8164 p^{41} T^{22} + p^{48} T^{24} \) | |
| 13 | \( ( 1 - 60 T + 104530 T^{2} + 2123716 T^{3} + 324849323 p T^{4} + 421717619768 T^{5} + 117361277505404 T^{6} + 421717619768 p^{4} T^{7} + 324849323 p^{9} T^{8} + 2123716 p^{12} T^{9} + 104530 p^{16} T^{10} - 60 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 17 | \( ( 1 - 324 T + 261922 T^{2} - 68620212 T^{3} + 41697749743 T^{4} - 8711801555976 T^{5} + 3978614951055580 T^{6} - 8711801555976 p^{4} T^{7} + 41697749743 p^{8} T^{8} - 68620212 p^{12} T^{9} + 261922 p^{16} T^{10} - 324 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 19 | \( 1 - 51844 p T^{2} + 482628291330 T^{4} - 154815161236503452 T^{6} + \)\(36\!\cdots\!71\)\( T^{8} - \)\(34\!\cdots\!28\)\( p T^{10} + \)\(95\!\cdots\!92\)\( T^{12} - \)\(34\!\cdots\!28\)\( p^{9} T^{14} + \)\(36\!\cdots\!71\)\( p^{16} T^{16} - 154815161236503452 p^{24} T^{18} + 482628291330 p^{32} T^{20} - 51844 p^{41} T^{22} + p^{48} T^{24} \) | |
| 23 | \( 1 - 2079052 T^{2} + 2071486470210 T^{4} - 1338585542516500508 T^{6} + \)\(63\!\cdots\!63\)\( T^{8} - \)\(24\!\cdots\!68\)\( T^{10} + \)\(74\!\cdots\!64\)\( T^{12} - \)\(24\!\cdots\!68\)\( p^{8} T^{14} + \)\(63\!\cdots\!63\)\( p^{16} T^{16} - 1338585542516500508 p^{24} T^{18} + 2071486470210 p^{32} T^{20} - 2079052 p^{40} T^{22} + p^{48} T^{24} \) | |
| 29 | \( ( 1 + 12 T + 2811618 T^{2} + 514568988 T^{3} + 3909907007055 T^{4} + 775020064717464 T^{5} + 3468892446953986012 T^{6} + 775020064717464 p^{4} T^{7} + 3909907007055 p^{8} T^{8} + 514568988 p^{12} T^{9} + 2811618 p^{16} T^{10} + 12 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 31 | \( 1 - 7463948 T^{2} + 25663893527106 T^{4} - 54273611234221848796 T^{6} + \)\(80\!\cdots\!31\)\( T^{8} - \)\(93\!\cdots\!56\)\( T^{10} + \)\(91\!\cdots\!80\)\( T^{12} - \)\(93\!\cdots\!56\)\( p^{8} T^{14} + \)\(80\!\cdots\!31\)\( p^{16} T^{16} - 54273611234221848796 p^{24} T^{18} + 25663893527106 p^{32} T^{20} - 7463948 p^{40} T^{22} + p^{48} T^{24} \) | |
| 37 | \( ( 1 + 212 T + 7019042 T^{2} + 1861141444 T^{3} + 24281538093071 T^{4} + 7754046867846696 T^{5} + 54940030635259586652 T^{6} + 7754046867846696 p^{4} T^{7} + 24281538093071 p^{8} T^{8} + 1861141444 p^{12} T^{9} + 7019042 p^{16} T^{10} + 212 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 41 | \( ( 1 + 3660 T + 13076226 T^{2} + 27884691228 T^{3} + 70033280174255 T^{4} + 127785009542971416 T^{5} + \)\(25\!\cdots\!96\)\( T^{6} + 127785009542971416 p^{4} T^{7} + 70033280174255 p^{8} T^{8} + 27884691228 p^{12} T^{9} + 13076226 p^{16} T^{10} + 3660 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 43 | \( 1 - 18040012 T^{2} + 167235315433986 T^{4} - \)\(10\!\cdots\!64\)\( T^{6} + \)\(57\!\cdots\!07\)\( T^{8} - \)\(25\!\cdots\!16\)\( T^{10} + \)\(51\!\cdots\!60\)\( p^{2} T^{12} - \)\(25\!\cdots\!16\)\( p^{8} T^{14} + \)\(57\!\cdots\!07\)\( p^{16} T^{16} - \)\(10\!\cdots\!64\)\( p^{24} T^{18} + 167235315433986 p^{32} T^{20} - 18040012 p^{40} T^{22} + p^{48} T^{24} \) | |
| 47 | \( 1 - 16773772 T^{2} + 106827665578050 T^{4} - \)\(34\!\cdots\!68\)\( T^{6} + \)\(16\!\cdots\!63\)\( T^{8} - \)\(17\!\cdots\!12\)\( T^{10} + \)\(11\!\cdots\!00\)\( T^{12} - \)\(17\!\cdots\!12\)\( p^{8} T^{14} + \)\(16\!\cdots\!63\)\( p^{16} T^{16} - \)\(34\!\cdots\!68\)\( p^{24} T^{18} + 106827665578050 p^{32} T^{20} - 16773772 p^{40} T^{22} + p^{48} T^{24} \) | |
| 53 | \( ( 1 - 3540 T + 35878946 T^{2} - 113381919300 T^{3} + 615516755990671 T^{4} - 1568317724977613736 T^{5} + \)\(62\!\cdots\!48\)\( T^{6} - 1568317724977613736 p^{4} T^{7} + 615516755990671 p^{8} T^{8} - 113381919300 p^{12} T^{9} + 35878946 p^{16} T^{10} - 3540 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 59 | \( 1 - 74505740 T^{2} + 2658112195198466 T^{4} - \)\(63\!\cdots\!12\)\( T^{6} + \)\(11\!\cdots\!35\)\( T^{8} - \)\(18\!\cdots\!84\)\( T^{10} + \)\(24\!\cdots\!92\)\( T^{12} - \)\(18\!\cdots\!84\)\( p^{8} T^{14} + \)\(11\!\cdots\!35\)\( p^{16} T^{16} - \)\(63\!\cdots\!12\)\( p^{24} T^{18} + 2658112195198466 p^{32} T^{20} - 74505740 p^{40} T^{22} + p^{48} T^{24} \) | |
| 61 | \( ( 1 - 4876 T + 47967954 T^{2} - 182589416108 T^{3} + 1144675304074223 T^{4} - 3759035837979994248 T^{5} + \)\(18\!\cdots\!64\)\( T^{6} - 3759035837979994248 p^{4} T^{7} + 1144675304074223 p^{8} T^{8} - 182589416108 p^{12} T^{9} + 47967954 p^{16} T^{10} - 4876 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 67 | \( 1 - 103320972 T^{2} + 6097515487665922 T^{4} - \)\(25\!\cdots\!00\)\( T^{6} + \)\(85\!\cdots\!83\)\( T^{8} - \)\(22\!\cdots\!20\)\( T^{10} + \)\(50\!\cdots\!28\)\( T^{12} - \)\(22\!\cdots\!20\)\( p^{8} T^{14} + \)\(85\!\cdots\!83\)\( p^{16} T^{16} - \)\(25\!\cdots\!00\)\( p^{24} T^{18} + 6097515487665922 p^{32} T^{20} - 103320972 p^{40} T^{22} + p^{48} T^{24} \) | |
| 71 | \( 1 - 184529420 T^{2} + 16662332162609346 T^{4} - \)\(99\!\cdots\!20\)\( T^{6} + \)\(43\!\cdots\!35\)\( T^{8} - \)\(15\!\cdots\!88\)\( T^{10} + \)\(42\!\cdots\!28\)\( T^{12} - \)\(15\!\cdots\!88\)\( p^{8} T^{14} + \)\(43\!\cdots\!35\)\( p^{16} T^{16} - \)\(99\!\cdots\!20\)\( p^{24} T^{18} + 16662332162609346 p^{32} T^{20} - 184529420 p^{40} T^{22} + p^{48} T^{24} \) | |
| 73 | \( ( 1 - 6876 T + 132960994 T^{2} - 643855190252 T^{3} + 7827237338728751 T^{4} - 30673892090460619192 T^{5} + \)\(28\!\cdots\!20\)\( T^{6} - 30673892090460619192 p^{4} T^{7} + 7827237338728751 p^{8} T^{8} - 643855190252 p^{12} T^{9} + 132960994 p^{16} T^{10} - 6876 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 79 | \( 1 - 248387724 T^{2} + 31997048356142658 T^{4} - \)\(28\!\cdots\!48\)\( T^{6} + \)\(19\!\cdots\!51\)\( T^{8} - \)\(10\!\cdots\!36\)\( T^{10} + \)\(43\!\cdots\!32\)\( T^{12} - \)\(10\!\cdots\!36\)\( p^{8} T^{14} + \)\(19\!\cdots\!51\)\( p^{16} T^{16} - \)\(28\!\cdots\!48\)\( p^{24} T^{18} + 31997048356142658 p^{32} T^{20} - 248387724 p^{40} T^{22} + p^{48} T^{24} \) | |
| 83 | \( 1 - 363703372 T^{2} + 66222999726752514 T^{4} - \)\(79\!\cdots\!40\)\( T^{6} + \)\(70\!\cdots\!07\)\( T^{8} - \)\(47\!\cdots\!20\)\( T^{10} + \)\(25\!\cdots\!64\)\( T^{12} - \)\(47\!\cdots\!20\)\( p^{8} T^{14} + \)\(70\!\cdots\!07\)\( p^{16} T^{16} - \)\(79\!\cdots\!40\)\( p^{24} T^{18} + 66222999726752514 p^{32} T^{20} - 363703372 p^{40} T^{22} + p^{48} T^{24} \) | |
| 89 | \( ( 1 - 6276 T + 167922466 T^{2} - 827882471412 T^{3} + 14021242065823151 T^{4} - 64307487192895729416 T^{5} + \)\(95\!\cdots\!44\)\( T^{6} - 64307487192895729416 p^{4} T^{7} + 14021242065823151 p^{8} T^{8} - 827882471412 p^{12} T^{9} + 167922466 p^{16} T^{10} - 6276 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
| 97 | \( ( 1 - 9852 T + 225718626 T^{2} - 1455507045644 T^{3} + 23966492834409327 T^{4} - \)\(10\!\cdots\!96\)\( T^{5} + \)\(21\!\cdots\!20\)\( T^{6} - \)\(10\!\cdots\!96\)\( p^{4} T^{7} + 23966492834409327 p^{8} T^{8} - 1455507045644 p^{12} T^{9} + 225718626 p^{16} T^{10} - 9852 p^{20} T^{11} + p^{24} T^{12} )^{2} \) | |
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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
−3.38994022301261021843102159891, −3.36196272229732135281252482988, −3.21293459053262471935722990577, −3.13112864024612997196605105601, −2.96351656000565546957808670988, −2.84959670043929433335081721597, −2.57145203630234394960183789489, −2.43959377454570953491614673165, −2.37778387114829817923574906662, −2.09539434249390964615005160308, −2.07338938764736738185273833474, −2.01463398518781606384631562863, −1.97054880891601940011325361801, −1.67430720295190526646834157798, −1.60683446675707987423497300980, −1.53082352267686719174261994318, −1.21336350577515901635994937641, −1.15087939207377376499518892955, −0.902170891729990257441183718514, −0.64296113099142959001918122785, −0.54590478709348524788014770173, −0.54184685405134808496517731521, −0.47984737506232935775831627763, −0.35311995484178642813515290878, −0.02253836115339392253801986333, 0.02253836115339392253801986333, 0.35311995484178642813515290878, 0.47984737506232935775831627763, 0.54184685405134808496517731521, 0.54590478709348524788014770173, 0.64296113099142959001918122785, 0.902170891729990257441183718514, 1.15087939207377376499518892955, 1.21336350577515901635994937641, 1.53082352267686719174261994318, 1.60683446675707987423497300980, 1.67430720295190526646834157798, 1.97054880891601940011325361801, 2.01463398518781606384631562863, 2.07338938764736738185273833474, 2.09539434249390964615005160308, 2.37778387114829817923574906662, 2.43959377454570953491614673165, 2.57145203630234394960183789489, 2.84959670043929433335081721597, 2.96351656000565546957808670988, 3.13112864024612997196605105601, 3.21293459053262471935722990577, 3.36196272229732135281252482988, 3.38994022301261021843102159891
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.