Dirichlet series
| L(s) = 1 | + 24·2-s + 11·3-s + 336·4-s + 150·5-s + 264·6-s + 366·7-s + 3.58e3·8-s − 219·9-s + 3.60e3·10-s + 151·11-s + 3.69e3·12-s + 463·13-s + 8.78e3·14-s + 1.65e3·15-s + 3.22e4·16-s + 644·17-s − 5.25e3·18-s + 3.43e3·19-s + 5.04e4·20-s + 4.02e3·21-s + 3.62e3·22-s + 3.17e3·23-s + 3.94e4·24-s + 1.31e4·25-s + 1.11e4·26-s − 4.86e3·27-s + 1.22e5·28-s + ⋯ |
| L(s) = 1 | + 4.24·2-s + 0.705·3-s + 21/2·4-s + 2.68·5-s + 2.99·6-s + 2.82·7-s + 19.7·8-s − 0.901·9-s + 11.3·10-s + 0.376·11-s + 7.40·12-s + 0.759·13-s + 11.9·14-s + 1.89·15-s + 63/2·16-s + 0.540·17-s − 3.82·18-s + 2.18·19-s + 28.1·20-s + 1.99·21-s + 1.59·22-s + 1.25·23-s + 13.9·24-s + 21/5·25-s + 3.22·26-s − 1.28·27-s + 29.6·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 5^{6} \cdot 23^{6}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 5^{6} \cdot 23^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.51959\times 10^{9}\) |
| Root analytic conductor: | \(6.07357\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{6} \cdot 5^{6} \cdot 23^{6} ,\ ( \ : [5/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(3060.606763\) |
| \(L(\frac12)\) | \(\approx\) | \(3060.606763\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{2} T )^{6} \) |
| 5 | \( ( 1 - p^{2} T )^{6} \) | |
| 23 | \( ( 1 - p^{2} T )^{6} \) | |
| good | 3 | \( 1 - 11 T + 340 T^{2} - 428 p T^{3} + 51350 T^{4} + 171289 p T^{5} + 786418 p^{2} T^{6} + 171289 p^{6} T^{7} + 51350 p^{10} T^{8} - 428 p^{16} T^{9} + 340 p^{20} T^{10} - 11 p^{25} T^{11} + p^{30} T^{12} \) |
| 7 | \( 1 - 366 T + 103374 T^{2} - 21636709 T^{3} + 3780851847 T^{4} - 563429762571 T^{5} + 78135637747060 T^{6} - 563429762571 p^{5} T^{7} + 3780851847 p^{10} T^{8} - 21636709 p^{15} T^{9} + 103374 p^{20} T^{10} - 366 p^{25} T^{11} + p^{30} T^{12} \) | |
| 11 | \( 1 - 151 T + 355317 T^{2} + 36935321 T^{3} + 62804104559 T^{4} + 11055411900860 T^{5} + 12309795813854966 T^{6} + 11055411900860 p^{5} T^{7} + 62804104559 p^{10} T^{8} + 36935321 p^{15} T^{9} + 355317 p^{20} T^{10} - 151 p^{25} T^{11} + p^{30} T^{12} \) | |
| 13 | \( 1 - 463 T + 1755420 T^{2} - 721123966 T^{3} + 1406313634410 T^{4} - 495239686373435 T^{5} + 663160057386526650 T^{6} - 495239686373435 p^{5} T^{7} + 1406313634410 p^{10} T^{8} - 721123966 p^{15} T^{9} + 1755420 p^{20} T^{10} - 463 p^{25} T^{11} + p^{30} T^{12} \) | |
| 17 | \( 1 - 644 T + 6317580 T^{2} - 4004299397 T^{3} + 19133784972609 T^{4} - 10686664059075051 T^{5} + 34314655060591793612 T^{6} - 10686664059075051 p^{5} T^{7} + 19133784972609 p^{10} T^{8} - 4004299397 p^{15} T^{9} + 6317580 p^{20} T^{10} - 644 p^{25} T^{11} + p^{30} T^{12} \) | |
| 19 | \( 1 - 3431 T + 14510995 T^{2} - 1789701805 p T^{3} + 87986010504367 T^{4} - 153570601636215932 T^{5} + \)\(28\!\cdots\!34\)\( T^{6} - 153570601636215932 p^{5} T^{7} + 87986010504367 p^{10} T^{8} - 1789701805 p^{16} T^{9} + 14510995 p^{20} T^{10} - 3431 p^{25} T^{11} + p^{30} T^{12} \) | |
| 29 | \( 1 - 5973 T + 91650533 T^{2} - 410194404296 T^{3} + 3620277801029051 T^{4} - 13283793875379225047 T^{5} + \)\(89\!\cdots\!30\)\( T^{6} - 13283793875379225047 p^{5} T^{7} + 3620277801029051 p^{10} T^{8} - 410194404296 p^{15} T^{9} + 91650533 p^{20} T^{10} - 5973 p^{25} T^{11} + p^{30} T^{12} \) | |
| 31 | \( 1 - 10262 T + 153058307 T^{2} - 1138089378083 T^{3} + 10121844242440725 T^{4} - 57569872910925864062 T^{5} + \)\(37\!\cdots\!38\)\( T^{6} - 57569872910925864062 p^{5} T^{7} + 10121844242440725 p^{10} T^{8} - 1138089378083 p^{15} T^{9} + 153058307 p^{20} T^{10} - 10262 p^{25} T^{11} + p^{30} T^{12} \) | |
| 37 | \( 1 - 17207 T + 431888448 T^{2} - 4755348314659 T^{3} + 68639886829794807 T^{4} - \)\(55\!\cdots\!46\)\( T^{5} + \)\(60\!\cdots\!84\)\( T^{6} - \)\(55\!\cdots\!46\)\( p^{5} T^{7} + 68639886829794807 p^{10} T^{8} - 4755348314659 p^{15} T^{9} + 431888448 p^{20} T^{10} - 17207 p^{25} T^{11} + p^{30} T^{12} \) | |
| 41 | \( 1 - 784 T + 253476033 T^{2} - 2467677244785 T^{3} + 43687623481670265 T^{4} - \)\(36\!\cdots\!06\)\( T^{5} + \)\(74\!\cdots\!52\)\( T^{6} - \)\(36\!\cdots\!06\)\( p^{5} T^{7} + 43687623481670265 p^{10} T^{8} - 2467677244785 p^{15} T^{9} + 253476033 p^{20} T^{10} - 784 p^{25} T^{11} + p^{30} T^{12} \) | |
| 43 | \( 1 - 13452 T + 217636746 T^{2} - 4759219236468 T^{3} + 55451668306308567 T^{4} - \)\(76\!\cdots\!72\)\( T^{5} + \)\(12\!\cdots\!68\)\( T^{6} - \)\(76\!\cdots\!72\)\( p^{5} T^{7} + 55451668306308567 p^{10} T^{8} - 4759219236468 p^{15} T^{9} + 217636746 p^{20} T^{10} - 13452 p^{25} T^{11} + p^{30} T^{12} \) | |
| 47 | \( 1 - 24572 T + 529127155 T^{2} - 7221945018998 T^{3} + 129054876865471775 T^{4} - \)\(24\!\cdots\!54\)\( T^{5} + \)\(44\!\cdots\!70\)\( T^{6} - \)\(24\!\cdots\!54\)\( p^{5} T^{7} + 129054876865471775 p^{10} T^{8} - 7221945018998 p^{15} T^{9} + 529127155 p^{20} T^{10} - 24572 p^{25} T^{11} + p^{30} T^{12} \) | |
| 53 | \( 1 - 17563 T + 1652487776 T^{2} - 16913530749691 T^{3} + 1191292404853862975 T^{4} - \)\(74\!\cdots\!50\)\( T^{5} + \)\(56\!\cdots\!48\)\( T^{6} - \)\(74\!\cdots\!50\)\( p^{5} T^{7} + 1191292404853862975 p^{10} T^{8} - 16913530749691 p^{15} T^{9} + 1652487776 p^{20} T^{10} - 17563 p^{25} T^{11} + p^{30} T^{12} \) | |
| 59 | \( 1 - 62911 T + 4484044024 T^{2} - 173684412463293 T^{3} + 7558467004179302415 T^{4} - \)\(22\!\cdots\!06\)\( T^{5} + \)\(71\!\cdots\!20\)\( T^{6} - \)\(22\!\cdots\!06\)\( p^{5} T^{7} + 7558467004179302415 p^{10} T^{8} - 173684412463293 p^{15} T^{9} + 4484044024 p^{20} T^{10} - 62911 p^{25} T^{11} + p^{30} T^{12} \) | |
| 61 | \( 1 - 32851 T + 3643301389 T^{2} - 59876900609149 T^{3} + 4606907749762337799 T^{4} - \)\(25\!\cdots\!72\)\( T^{5} + \)\(37\!\cdots\!46\)\( T^{6} - \)\(25\!\cdots\!72\)\( p^{5} T^{7} + 4606907749762337799 p^{10} T^{8} - 59876900609149 p^{15} T^{9} + 3643301389 p^{20} T^{10} - 32851 p^{25} T^{11} + p^{30} T^{12} \) | |
| 67 | \( 1 - 54177 T + 6108810386 T^{2} - 178322777623147 T^{3} + 12428438900237991015 T^{4} - \)\(19\!\cdots\!02\)\( T^{5} + \)\(16\!\cdots\!08\)\( T^{6} - \)\(19\!\cdots\!02\)\( p^{5} T^{7} + 12428438900237991015 p^{10} T^{8} - 178322777623147 p^{15} T^{9} + 6108810386 p^{20} T^{10} - 54177 p^{25} T^{11} + p^{30} T^{12} \) | |
| 71 | \( 1 + 14368 T + 4786620187 T^{2} + 11303137706613 T^{3} + 10096952701930389841 T^{4} - \)\(18\!\cdots\!56\)\( T^{5} + \)\(16\!\cdots\!62\)\( T^{6} - \)\(18\!\cdots\!56\)\( p^{5} T^{7} + 10096952701930389841 p^{10} T^{8} + 11303137706613 p^{15} T^{9} + 4786620187 p^{20} T^{10} + 14368 p^{25} T^{11} + p^{30} T^{12} \) | |
| 73 | \( 1 - 33276 T + 5436660699 T^{2} - 100619097429566 T^{3} + 9071495432376681579 T^{4} - \)\(52\!\cdots\!02\)\( T^{5} + \)\(95\!\cdots\!82\)\( T^{6} - \)\(52\!\cdots\!02\)\( p^{5} T^{7} + 9071495432376681579 p^{10} T^{8} - 100619097429566 p^{15} T^{9} + 5436660699 p^{20} T^{10} - 33276 p^{25} T^{11} + p^{30} T^{12} \) | |
| 79 | \( 1 - 74296 T + 9374051830 T^{2} - 857488632428152 T^{3} + 61980406647937510911 T^{4} - \)\(38\!\cdots\!08\)\( T^{5} + \)\(26\!\cdots\!36\)\( T^{6} - \)\(38\!\cdots\!08\)\( p^{5} T^{7} + 61980406647937510911 p^{10} T^{8} - 857488632428152 p^{15} T^{9} + 9374051830 p^{20} T^{10} - 74296 p^{25} T^{11} + p^{30} T^{12} \) | |
| 83 | \( 1 - 65145 T + 15698306438 T^{2} - 822839489877131 T^{3} + \)\(11\!\cdots\!95\)\( T^{4} - \)\(53\!\cdots\!30\)\( T^{5} + \)\(56\!\cdots\!36\)\( T^{6} - \)\(53\!\cdots\!30\)\( p^{5} T^{7} + \)\(11\!\cdots\!95\)\( p^{10} T^{8} - 822839489877131 p^{15} T^{9} + 15698306438 p^{20} T^{10} - 65145 p^{25} T^{11} + p^{30} T^{12} \) | |
| 89 | \( 1 + 67562 T + 20147599990 T^{2} + 529916083941714 T^{3} + \)\(14\!\cdots\!19\)\( T^{4} - \)\(98\!\cdots\!68\)\( T^{5} + \)\(71\!\cdots\!00\)\( T^{6} - \)\(98\!\cdots\!68\)\( p^{5} T^{7} + \)\(14\!\cdots\!19\)\( p^{10} T^{8} + 529916083941714 p^{15} T^{9} + 20147599990 p^{20} T^{10} + 67562 p^{25} T^{11} + p^{30} T^{12} \) | |
| 97 | \( 1 + 13201 T + 36842583391 T^{2} + 264953503082617 T^{3} + \)\(65\!\cdots\!19\)\( T^{4} + \)\(34\!\cdots\!42\)\( T^{5} + \)\(71\!\cdots\!10\)\( T^{6} + \)\(34\!\cdots\!42\)\( p^{5} T^{7} + \)\(65\!\cdots\!19\)\( p^{10} T^{8} + 264953503082617 p^{15} T^{9} + 36842583391 p^{20} T^{10} + 13201 p^{25} T^{11} + p^{30} T^{12} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.74552525207650653638051228740, −5.30294291223524876694027448388, −5.15529317473857079762475398693, −5.12556191437139820811933555264, −5.06880824691811950536914761036, −4.98126041627884667342363879708, −4.67783831485303055492909807627, −4.59748640701370504457675174118, −4.13798157349183856005171400386, −3.87598362180098786679798796579, −3.73482481996965186381579605723, −3.64064138480563134386134311583, −3.48854632119274997277390113189, −2.82069548758850593552509454282, −2.78985778473477628799191337104, −2.56690324299940000517972013403, −2.47010475022568671553519075335, −2.45963912696429429799603176370, −2.25682569205694003898085169941, −1.61993899503898064039059172109, −1.45121459314454758111087132877, −1.40912632619176340353584002858, −0.968438749829031031849432693166, −0.944271919126649588585626030990, −0.858804203775813328155034462990, 0.858804203775813328155034462990, 0.944271919126649588585626030990, 0.968438749829031031849432693166, 1.40912632619176340353584002858, 1.45121459314454758111087132877, 1.61993899503898064039059172109, 2.25682569205694003898085169941, 2.45963912696429429799603176370, 2.47010475022568671553519075335, 2.56690324299940000517972013403, 2.78985778473477628799191337104, 2.82069548758850593552509454282, 3.48854632119274997277390113189, 3.64064138480563134386134311583, 3.73482481996965186381579605723, 3.87598362180098786679798796579, 4.13798157349183856005171400386, 4.59748640701370504457675174118, 4.67783831485303055492909807627, 4.98126041627884667342363879708, 5.06880824691811950536914761036, 5.12556191437139820811933555264, 5.15529317473857079762475398693, 5.30294291223524876694027448388, 5.74552525207650653638051228740
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.