Dirichlet series
| L(s) = 1 | + 3·2-s + 474·4-s − 1.98e3·5-s + 3.69e3·7-s + 5.43e3·8-s − 5.94e3·10-s − 1.68e4·11-s − 1.16e5·13-s + 1.10e4·14-s + 2.60e5·16-s + 2.02e6·17-s − 3.04e4·19-s − 9.39e5·20-s − 5.05e4·22-s − 2.92e6·23-s + 3.82e6·25-s − 3.50e5·26-s + 1.75e6·28-s − 5.76e6·29-s + 6.57e6·31-s + 3.64e6·32-s + 6.08e6·34-s − 7.32e6·35-s − 2.33e7·37-s − 9.13e4·38-s − 1.07e7·40-s + 2.22e7·41-s + ⋯ |
| L(s) = 1 | + 0.132·2-s + 0.925·4-s − 1.41·5-s + 0.581·7-s + 0.469·8-s − 0.188·10-s − 0.347·11-s − 1.13·13-s + 0.0770·14-s + 0.994·16-s + 5.88·17-s − 0.0535·19-s − 1.31·20-s − 0.0460·22-s − 2.18·23-s + 1.96·25-s − 0.150·26-s + 0.538·28-s − 1.51·29-s + 1.27·31-s + 0.615·32-s + 0.780·34-s − 0.824·35-s − 2.05·37-s − 0.00710·38-s − 0.666·40-s + 1.22·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(3^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.27150\times 10^{9}\) |
| Root analytic conductor: | \(6.45893\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 3^{24} ,\ ( \ : [9/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(16.12395404\) |
| \(L(\frac12)\) | \(\approx\) | \(16.12395404\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 3 T - 465 T^{2} - 1311 p T^{3} - 4053 p^{2} T^{4} + 59631 p^{4} T^{5} + 2365657 p^{6} T^{6} + 59631 p^{13} T^{7} - 4053 p^{20} T^{8} - 1311 p^{28} T^{9} - 465 p^{36} T^{10} - 3 p^{45} T^{11} + p^{54} T^{12} \) |
| 5 | \( 1 + 1983 T + 103323 T^{2} - 247551972 T^{3} - 591216811419 T^{4} - 283387543520379 p^{2} T^{5} - 21385798046547314 p^{4} T^{6} - 283387543520379 p^{11} T^{7} - 591216811419 p^{18} T^{8} - 247551972 p^{27} T^{9} + 103323 p^{36} T^{10} + 1983 p^{45} T^{11} + p^{54} T^{12} \) | |
| 7 | \( 1 - 3693 T - 44244555 T^{2} + 80438061026 p T^{3} - 8566798580577 p^{2} T^{4} - 32781753343125999 p^{3} T^{5} + 52011583402657501302 p^{4} T^{6} - 32781753343125999 p^{12} T^{7} - 8566798580577 p^{20} T^{8} + 80438061026 p^{28} T^{9} - 44244555 p^{36} T^{10} - 3693 p^{45} T^{11} + p^{54} T^{12} \) | |
| 11 | \( 1 + 1533 p T - 4293537387 T^{2} - 142744081739034 T^{3} + 8308273094930256387 T^{4} + \)\(23\!\cdots\!43\)\( T^{5} - \)\(12\!\cdots\!62\)\( T^{6} + \)\(23\!\cdots\!43\)\( p^{9} T^{7} + 8308273094930256387 p^{18} T^{8} - 142744081739034 p^{27} T^{9} - 4293537387 p^{36} T^{10} + 1533 p^{46} T^{11} + p^{54} T^{12} \) | |
| 13 | \( 1 + 116916 T - 1370360679 T^{2} - 99157858675444 p T^{3} - \)\(11\!\cdots\!86\)\( T^{4} - \)\(12\!\cdots\!36\)\( T^{5} + \)\(77\!\cdots\!41\)\( T^{6} - \)\(12\!\cdots\!36\)\( p^{9} T^{7} - \)\(11\!\cdots\!86\)\( p^{18} T^{8} - 99157858675444 p^{28} T^{9} - 1370360679 p^{36} T^{10} + 116916 p^{45} T^{11} + p^{54} T^{12} \) | |
| 17 | \( ( 1 - 1014048 T + 618168912243 T^{2} - 14610762415245888 p T^{3} + 618168912243 p^{9} T^{4} - 1014048 p^{18} T^{5} + p^{27} T^{6} )^{2} \) | |
| 19 | \( ( 1 + 15222 T + 368176140789 T^{2} + 14286898979746540 T^{3} + 368176140789 p^{9} T^{4} + 15222 p^{18} T^{5} + p^{27} T^{6} )^{2} \) | |
| 23 | \( 1 + 127266 p T + 4099418030739 T^{2} + 1130353484234590038 T^{3} - \)\(55\!\cdots\!58\)\( T^{4} - \)\(10\!\cdots\!78\)\( T^{5} - \)\(16\!\cdots\!21\)\( T^{6} - \)\(10\!\cdots\!78\)\( p^{9} T^{7} - \)\(55\!\cdots\!58\)\( p^{18} T^{8} + 1130353484234590038 p^{27} T^{9} + 4099418030739 p^{36} T^{10} + 127266 p^{46} T^{11} + p^{54} T^{12} \) | |
| 29 | \( 1 + 5768790 T - 13979851684719 T^{2} - 53542672936804004070 T^{3} + \)\(60\!\cdots\!50\)\( T^{4} + \)\(10\!\cdots\!70\)\( T^{5} - \)\(67\!\cdots\!91\)\( T^{6} + \)\(10\!\cdots\!70\)\( p^{9} T^{7} + \)\(60\!\cdots\!50\)\( p^{18} T^{8} - 53542672936804004070 p^{27} T^{9} - 13979851684719 p^{36} T^{10} + 5768790 p^{45} T^{11} + p^{54} T^{12} \) | |
| 31 | \( 1 - 6575223 T - 18335259491091 T^{2} + \)\(16\!\cdots\!10\)\( T^{3} + \)\(28\!\cdots\!75\)\( T^{4} - \)\(74\!\cdots\!23\)\( T^{5} - \)\(12\!\cdots\!42\)\( T^{6} - \)\(74\!\cdots\!23\)\( p^{9} T^{7} + \)\(28\!\cdots\!75\)\( p^{18} T^{8} + \)\(16\!\cdots\!10\)\( p^{27} T^{9} - 18335259491091 p^{36} T^{10} - 6575223 p^{45} T^{11} + p^{54} T^{12} \) | |
| 37 | \( ( 1 + 11686026 T + 301605126050715 T^{2} + \)\(19\!\cdots\!24\)\( T^{3} + 301605126050715 p^{9} T^{4} + 11686026 p^{18} T^{5} + p^{27} T^{6} )^{2} \) | |
| 41 | \( 1 - 22213518 T - 362008317528627 T^{2} + \)\(27\!\cdots\!14\)\( T^{3} + \)\(20\!\cdots\!02\)\( T^{4} + \)\(74\!\cdots\!02\)\( T^{5} - \)\(10\!\cdots\!47\)\( T^{6} + \)\(74\!\cdots\!02\)\( p^{9} T^{7} + \)\(20\!\cdots\!02\)\( p^{18} T^{8} + \)\(27\!\cdots\!14\)\( p^{27} T^{9} - 362008317528627 p^{36} T^{10} - 22213518 p^{45} T^{11} + p^{54} T^{12} \) | |
| 43 | \( 1 + 45384414 T + 1474961716068231 T^{2} + \)\(26\!\cdots\!22\)\( T^{3} + \)\(51\!\cdots\!14\)\( T^{4} - \)\(15\!\cdots\!14\)\( T^{5} - \)\(43\!\cdots\!29\)\( T^{6} - \)\(15\!\cdots\!14\)\( p^{9} T^{7} + \)\(51\!\cdots\!14\)\( p^{18} T^{8} + \)\(26\!\cdots\!22\)\( p^{27} T^{9} + 1474961716068231 p^{36} T^{10} + 45384414 p^{45} T^{11} + p^{54} T^{12} \) | |
| 47 | \( 1 + 12392034 T - 2538046295995317 T^{2} - \)\(15\!\cdots\!10\)\( T^{3} + \)\(40\!\cdots\!74\)\( T^{4} - \)\(12\!\cdots\!54\)\( T^{5} - \)\(51\!\cdots\!85\)\( T^{6} - \)\(12\!\cdots\!54\)\( p^{9} T^{7} + \)\(40\!\cdots\!74\)\( p^{18} T^{8} - \)\(15\!\cdots\!10\)\( p^{27} T^{9} - 2538046295995317 p^{36} T^{10} + 12392034 p^{45} T^{11} + p^{54} T^{12} \) | |
| 53 | \( ( 1 - 80579637 T + 8930124888804942 T^{2} - \)\(39\!\cdots\!45\)\( T^{3} + 8930124888804942 p^{9} T^{4} - 80579637 p^{18} T^{5} + p^{27} T^{6} )^{2} \) | |
| 59 | \( 1 + 244026660 T + 22761229159237071 T^{2} + \)\(15\!\cdots\!60\)\( T^{3} + \)\(13\!\cdots\!10\)\( T^{4} - \)\(11\!\cdots\!00\)\( T^{5} - \)\(12\!\cdots\!61\)\( T^{6} - \)\(11\!\cdots\!00\)\( p^{9} T^{7} + \)\(13\!\cdots\!10\)\( p^{18} T^{8} + \)\(15\!\cdots\!60\)\( p^{27} T^{9} + 22761229159237071 p^{36} T^{10} + 244026660 p^{45} T^{11} + p^{54} T^{12} \) | |
| 61 | \( 1 + 369729960 T + 61176699041773161 T^{2} + \)\(83\!\cdots\!04\)\( T^{3} + \)\(11\!\cdots\!42\)\( T^{4} + \)\(13\!\cdots\!32\)\( T^{5} + \)\(13\!\cdots\!05\)\( T^{6} + \)\(13\!\cdots\!32\)\( p^{9} T^{7} + \)\(11\!\cdots\!42\)\( p^{18} T^{8} + \)\(83\!\cdots\!04\)\( p^{27} T^{9} + 61176699041773161 p^{36} T^{10} + 369729960 p^{45} T^{11} + p^{54} T^{12} \) | |
| 67 | \( 1 - 252614586 T - 20146902594742977 T^{2} + \)\(24\!\cdots\!90\)\( T^{3} + \)\(17\!\cdots\!74\)\( T^{4} - \)\(26\!\cdots\!74\)\( T^{5} - \)\(58\!\cdots\!25\)\( T^{6} - \)\(26\!\cdots\!74\)\( p^{9} T^{7} + \)\(17\!\cdots\!74\)\( p^{18} T^{8} + \)\(24\!\cdots\!90\)\( p^{27} T^{9} - 20146902594742977 p^{36} T^{10} - 252614586 p^{45} T^{11} + p^{54} T^{12} \) | |
| 71 | \( ( 1 - 403193088 T + 169497868047009621 T^{2} - \)\(35\!\cdots\!16\)\( T^{3} + 169497868047009621 p^{9} T^{4} - 403193088 p^{18} T^{5} + p^{27} T^{6} )^{2} \) | |
| 73 | \( ( 1 + 5570229 p T + 211748488417949982 T^{2} + \)\(47\!\cdots\!97\)\( T^{3} + 211748488417949982 p^{9} T^{4} + 5570229 p^{19} T^{5} + p^{27} T^{6} )^{2} \) | |
| 79 | \( 1 + 265451856 T - 121698022618556205 T^{2} - \)\(59\!\cdots\!44\)\( T^{3} - \)\(78\!\cdots\!18\)\( T^{4} + \)\(22\!\cdots\!88\)\( T^{5} + \)\(10\!\cdots\!83\)\( T^{6} + \)\(22\!\cdots\!88\)\( p^{9} T^{7} - \)\(78\!\cdots\!18\)\( p^{18} T^{8} - \)\(59\!\cdots\!44\)\( p^{27} T^{9} - 121698022618556205 p^{36} T^{10} + 265451856 p^{45} T^{11} + p^{54} T^{12} \) | |
| 83 | \( 1 + 121625871 T - 506898598091368515 T^{2} - \)\(23\!\cdots\!62\)\( T^{3} + \)\(16\!\cdots\!19\)\( T^{4} + \)\(31\!\cdots\!47\)\( T^{5} - \)\(36\!\cdots\!62\)\( T^{6} + \)\(31\!\cdots\!47\)\( p^{9} T^{7} + \)\(16\!\cdots\!19\)\( p^{18} T^{8} - \)\(23\!\cdots\!62\)\( p^{27} T^{9} - 506898598091368515 p^{36} T^{10} + 121625871 p^{45} T^{11} + p^{54} T^{12} \) | |
| 89 | \( ( 1 - 377904006 T + 336010266400343559 T^{2} - \)\(36\!\cdots\!08\)\( T^{3} + 336010266400343559 p^{9} T^{4} - 377904006 p^{18} T^{5} + p^{27} T^{6} )^{2} \) | |
| 97 | \( 1 - 438907539 T - 1957114181579258589 T^{2} + \)\(33\!\cdots\!52\)\( T^{3} + \)\(27\!\cdots\!61\)\( T^{4} - \)\(19\!\cdots\!21\)\( T^{5} - \)\(23\!\cdots\!02\)\( T^{6} - \)\(19\!\cdots\!21\)\( p^{9} T^{7} + \)\(27\!\cdots\!61\)\( p^{18} T^{8} + \)\(33\!\cdots\!52\)\( p^{27} T^{9} - 1957114181579258589 p^{36} T^{10} - 438907539 p^{45} T^{11} + p^{54} T^{12} \) | |
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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
−6.17335988599156740565729289299, −5.98382788295274525885532946983, −5.68610545153747466597590874781, −5.46469723066502006115109606732, −5.13212776663107676032529576462, −5.12687328243705742334798788586, −5.09383226598210366731435037352, −4.36135092702966919326975063530, −4.32172249335086621089573416215, −4.22769010719140434162750189650, −3.76770847262583606637455834903, −3.36116264280635447140444799933, −3.34522235015321195888823863944, −3.11111356171392011906069762585, −3.11089624055602592711906574142, −2.86905019722908484823216701433, −2.16323991930885370903791831891, −1.94704111393420044280746946576, −1.84134361522048391132144640682, −1.58001663749406013424656422459, −1.31236360864619826941037793309, −0.800586134874715879831539781135, −0.73866229231119596477139201001, −0.57203284448338009982210593920, −0.32106350409031836135293643095, 0.32106350409031836135293643095, 0.57203284448338009982210593920, 0.73866229231119596477139201001, 0.800586134874715879831539781135, 1.31236360864619826941037793309, 1.58001663749406013424656422459, 1.84134361522048391132144640682, 1.94704111393420044280746946576, 2.16323991930885370903791831891, 2.86905019722908484823216701433, 3.11089624055602592711906574142, 3.11111356171392011906069762585, 3.34522235015321195888823863944, 3.36116264280635447140444799933, 3.76770847262583606637455834903, 4.22769010719140434162750189650, 4.32172249335086621089573416215, 4.36135092702966919326975063530, 5.09383226598210366731435037352, 5.12687328243705742334798788586, 5.13212776663107676032529576462, 5.46469723066502006115109606732, 5.68610545153747466597590874781, 5.98382788295274525885532946983, 6.17335988599156740565729289299
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.