Dirichlet series
| L(s) = 1 | + 192·2-s + 296·3-s + 1.84e4·4-s + 1.44e4·5-s + 5.68e4·6-s − 3.22e5·7-s + 1.04e6·8-s + 4.38e4·9-s + 2.77e6·10-s + 1.65e6·11-s + 5.45e6·12-s − 4.64e6·13-s − 6.18e7·14-s + 4.28e6·15-s + 2.51e7·16-s + 5.12e7·17-s + 8.41e6·18-s + 2.66e8·20-s − 9.53e7·21-s + 3.17e8·22-s + 1.05e8·23-s + 3.10e8·24-s + 2.37e7·25-s − 8.92e8·26-s + 2.65e8·27-s − 5.93e9·28-s + 8.21e8·30-s + ⋯ |
| L(s) = 1 | + 3·2-s + 0.406·3-s + 9/2·4-s + 0.925·5-s + 1.21·6-s − 2.73·7-s + 4·8-s + 0.0824·9-s + 2.77·10-s + 0.932·11-s + 1.82·12-s − 0.962·13-s − 8.21·14-s + 0.375·15-s + 3/2·16-s + 2.12·17-s + 0.247·18-s + 4.16·20-s − 1.11·21-s + 2.79·22-s + 0.714·23-s + 1.62·24-s + 0.0974·25-s − 2.88·26-s + 0.686·27-s − 12.3·28-s + 1.12·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000000 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(13-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1000000 ^{s/2} \, \Gamma_{\C}(s+6)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(1000000\) = \(2^{6} \cdot 5^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(582987.\) |
| Root analytic conductor: | \(3.02323\) |
| Motivic weight: | \(12\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 1000000,\ (\ :[6]^{6}),\ 1)\) |
Particular Values
| \(L(\frac{13}{2})\) | \(\approx\) | \(20.89037056\) |
| \(L(\frac12)\) | \(\approx\) | \(20.89037056\) |
| \(L(7)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{6} T + p^{11} T^{2} )^{3} \) |
| 5 | \( 1 - 2892 p T + 296487 p^{4} T^{2} + 2989032 p^{8} T^{3} + 296487 p^{16} T^{4} - 2892 p^{25} T^{5} + p^{36} T^{6} \) | |
| good | 3 | \( 1 - 296 T + 43808 T^{2} - 3281168 p^{4} T^{3} - 43045937 p^{8} T^{4} + 11435611912 p^{9} T^{5} - 320670583072 p^{10} T^{6} + 11435611912 p^{21} T^{7} - 43045937 p^{32} T^{8} - 3281168 p^{40} T^{9} + 43808 p^{48} T^{10} - 296 p^{60} T^{11} + p^{72} T^{12} \) |
| 7 | \( 1 + 322104 T + 51875493408 T^{2} + 6387501326203232 T^{3} + \)\(31\!\cdots\!43\)\( T^{4} - \)\(43\!\cdots\!32\)\( p T^{5} - \)\(11\!\cdots\!72\)\( p^{2} T^{6} - \)\(43\!\cdots\!32\)\( p^{13} T^{7} + \)\(31\!\cdots\!43\)\( p^{24} T^{8} + 6387501326203232 p^{36} T^{9} + 51875493408 p^{48} T^{10} + 322104 p^{60} T^{11} + p^{72} T^{12} \) | |
| 11 | \( ( 1 - 826356 T + 548985916725 p T^{2} - 56567159371396360 p^{2} T^{3} + 548985916725 p^{13} T^{4} - 826356 p^{24} T^{5} + p^{36} T^{6} )^{2} \) | |
| 13 | \( 1 + 4646814 T + 10796440175298 T^{2} + 59958496622356841022 T^{3} + \)\(87\!\cdots\!23\)\( T^{4} + \)\(44\!\cdots\!76\)\( T^{5} + \)\(13\!\cdots\!52\)\( T^{6} + \)\(44\!\cdots\!76\)\( p^{12} T^{7} + \)\(87\!\cdots\!23\)\( p^{24} T^{8} + 59958496622356841022 p^{36} T^{9} + 10796440175298 p^{48} T^{10} + 4646814 p^{60} T^{11} + p^{72} T^{12} \) | |
| 17 | \( 1 - 3011778 p T + 4535403360642 p^{2} T^{2} - \)\(34\!\cdots\!38\)\( T^{3} + \)\(42\!\cdots\!03\)\( T^{4} + \)\(32\!\cdots\!76\)\( T^{5} - \)\(10\!\cdots\!68\)\( T^{6} + \)\(32\!\cdots\!76\)\( p^{12} T^{7} + \)\(42\!\cdots\!03\)\( p^{24} T^{8} - \)\(34\!\cdots\!38\)\( p^{36} T^{9} + 4535403360642 p^{50} T^{10} - 3011778 p^{61} T^{11} + p^{72} T^{12} \) | |
| 19 | \( 1 - 3019774946894166 T^{2} - \)\(27\!\cdots\!85\)\( T^{4} + \)\(12\!\cdots\!80\)\( T^{6} - \)\(27\!\cdots\!85\)\( p^{24} T^{8} - 3019774946894166 p^{48} T^{10} + p^{72} T^{12} \) | |
| 23 | \( 1 - 105826896 T + 5599665958497408 T^{2} - \)\(16\!\cdots\!56\)\( p T^{3} + \)\(94\!\cdots\!43\)\( T^{4} - \)\(70\!\cdots\!64\)\( T^{5} + \)\(90\!\cdots\!72\)\( T^{6} - \)\(70\!\cdots\!64\)\( p^{12} T^{7} + \)\(94\!\cdots\!43\)\( p^{24} T^{8} - \)\(16\!\cdots\!56\)\( p^{37} T^{9} + 5599665958497408 p^{48} T^{10} - 105826896 p^{60} T^{11} + p^{72} T^{12} \) | |
| 29 | \( 1 - 887923775060223846 T^{2} + \)\(42\!\cdots\!15\)\( T^{4} - \)\(15\!\cdots\!20\)\( T^{6} + \)\(42\!\cdots\!15\)\( p^{24} T^{8} - 887923775060223846 p^{48} T^{10} + p^{72} T^{12} \) | |
| 31 | \( ( 1 + 1333558584 T + 1976764301333685735 T^{2} + \)\(20\!\cdots\!00\)\( T^{3} + 1976764301333685735 p^{12} T^{4} + 1333558584 p^{24} T^{5} + p^{36} T^{6} )^{2} \) | |
| 37 | \( 1 - 1747956246 T + 1527675518965206258 T^{2} - \)\(11\!\cdots\!98\)\( T^{3} + \)\(13\!\cdots\!43\)\( T^{4} - \)\(15\!\cdots\!84\)\( T^{5} + \)\(13\!\cdots\!72\)\( T^{6} - \)\(15\!\cdots\!84\)\( p^{12} T^{7} + \)\(13\!\cdots\!43\)\( p^{24} T^{8} - \)\(11\!\cdots\!98\)\( p^{36} T^{9} + 1527675518965206258 p^{48} T^{10} - 1747956246 p^{60} T^{11} + p^{72} T^{12} \) | |
| 41 | \( ( 1 - 11361049116 T + 97304369679810531495 T^{2} - \)\(53\!\cdots\!40\)\( T^{3} + 97304369679810531495 p^{12} T^{4} - 11361049116 p^{24} T^{5} + p^{36} T^{6} )^{2} \) | |
| 43 | \( 1 + 15890524824 T + \)\(12\!\cdots\!88\)\( T^{2} + \)\(10\!\cdots\!72\)\( T^{3} + \)\(97\!\cdots\!03\)\( T^{4} + \)\(68\!\cdots\!96\)\( T^{5} + \)\(40\!\cdots\!32\)\( T^{6} + \)\(68\!\cdots\!96\)\( p^{12} T^{7} + \)\(97\!\cdots\!03\)\( p^{24} T^{8} + \)\(10\!\cdots\!72\)\( p^{36} T^{9} + \)\(12\!\cdots\!88\)\( p^{48} T^{10} + 15890524824 p^{60} T^{11} + p^{72} T^{12} \) | |
| 47 | \( 1 + 18495531264 T + \)\(17\!\cdots\!48\)\( T^{2} + \)\(25\!\cdots\!12\)\( T^{3} + \)\(43\!\cdots\!23\)\( T^{4} + \)\(39\!\cdots\!56\)\( T^{5} + \)\(32\!\cdots\!52\)\( T^{6} + \)\(39\!\cdots\!56\)\( p^{12} T^{7} + \)\(43\!\cdots\!23\)\( p^{24} T^{8} + \)\(25\!\cdots\!12\)\( p^{36} T^{9} + \)\(17\!\cdots\!48\)\( p^{48} T^{10} + 18495531264 p^{60} T^{11} + p^{72} T^{12} \) | |
| 53 | \( 1 + 88020413514 T + \)\(38\!\cdots\!98\)\( T^{2} + \)\(12\!\cdots\!62\)\( T^{3} + \)\(33\!\cdots\!23\)\( T^{4} + \)\(85\!\cdots\!56\)\( T^{5} + \)\(20\!\cdots\!52\)\( T^{6} + \)\(85\!\cdots\!56\)\( p^{12} T^{7} + \)\(33\!\cdots\!23\)\( p^{24} T^{8} + \)\(12\!\cdots\!62\)\( p^{36} T^{9} + \)\(38\!\cdots\!98\)\( p^{48} T^{10} + 88020413514 p^{60} T^{11} + p^{72} T^{12} \) | |
| 59 | \( 1 - \)\(38\!\cdots\!86\)\( T^{2} + \)\(11\!\cdots\!15\)\( T^{4} - \)\(24\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!15\)\( p^{24} T^{8} - \)\(38\!\cdots\!86\)\( p^{48} T^{10} + p^{72} T^{12} \) | |
| 61 | \( ( 1 - 145897445676 T + \)\(13\!\cdots\!55\)\( T^{2} - \)\(76\!\cdots\!80\)\( T^{3} + \)\(13\!\cdots\!55\)\( p^{12} T^{4} - 145897445676 p^{24} T^{5} + p^{36} T^{6} )^{2} \) | |
| 67 | \( 1 + 3887251464 T + 7555361972185071648 T^{2} + \)\(14\!\cdots\!92\)\( T^{3} - \)\(56\!\cdots\!77\)\( T^{4} - \)\(75\!\cdots\!84\)\( T^{5} + \)\(10\!\cdots\!52\)\( T^{6} - \)\(75\!\cdots\!84\)\( p^{12} T^{7} - \)\(56\!\cdots\!77\)\( p^{24} T^{8} + \)\(14\!\cdots\!92\)\( p^{36} T^{9} + 7555361972185071648 p^{48} T^{10} + 3887251464 p^{60} T^{11} + p^{72} T^{12} \) | |
| 71 | \( ( 1 - 464567507736 T + \)\(11\!\cdots\!55\)\( T^{2} - \)\(18\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!55\)\( p^{12} T^{4} - 464567507736 p^{24} T^{5} + p^{36} T^{6} )^{2} \) | |
| 73 | \( 1 - 12678070086 T + 80366730552764023698 T^{2} - \)\(64\!\cdots\!18\)\( T^{3} + \)\(92\!\cdots\!23\)\( T^{4} - \)\(18\!\cdots\!04\)\( T^{5} + \)\(16\!\cdots\!52\)\( T^{6} - \)\(18\!\cdots\!04\)\( p^{12} T^{7} + \)\(92\!\cdots\!23\)\( p^{24} T^{8} - \)\(64\!\cdots\!18\)\( p^{36} T^{9} + 80366730552764023698 p^{48} T^{10} - 12678070086 p^{60} T^{11} + p^{72} T^{12} \) | |
| 79 | \( 1 - \)\(11\!\cdots\!46\)\( T^{2} + \)\(39\!\cdots\!15\)\( T^{4} - \)\(19\!\cdots\!20\)\( T^{6} + \)\(39\!\cdots\!15\)\( p^{24} T^{8} - \)\(11\!\cdots\!46\)\( p^{48} T^{10} + p^{72} T^{12} \) | |
| 83 | \( 1 - 68676615456 T + \)\(23\!\cdots\!68\)\( T^{2} + \)\(31\!\cdots\!52\)\( T^{3} - \)\(11\!\cdots\!37\)\( T^{4} - \)\(21\!\cdots\!04\)\( T^{5} + \)\(67\!\cdots\!92\)\( T^{6} - \)\(21\!\cdots\!04\)\( p^{12} T^{7} - \)\(11\!\cdots\!37\)\( p^{24} T^{8} + \)\(31\!\cdots\!52\)\( p^{36} T^{9} + \)\(23\!\cdots\!68\)\( p^{48} T^{10} - 68676615456 p^{60} T^{11} + p^{72} T^{12} \) | |
| 89 | \( 1 - \)\(12\!\cdots\!26\)\( T^{2} + \)\(65\!\cdots\!15\)\( T^{4} - \)\(20\!\cdots\!20\)\( T^{6} + \)\(65\!\cdots\!15\)\( p^{24} T^{8} - \)\(12\!\cdots\!26\)\( p^{48} T^{10} + p^{72} T^{12} \) | |
| 97 | \( 1 - 675735777846 T + \)\(22\!\cdots\!58\)\( T^{2} - \)\(35\!\cdots\!58\)\( T^{3} - \)\(45\!\cdots\!57\)\( T^{4} + \)\(57\!\cdots\!96\)\( T^{5} - \)\(21\!\cdots\!28\)\( T^{6} + \)\(57\!\cdots\!96\)\( p^{12} T^{7} - \)\(45\!\cdots\!57\)\( p^{24} T^{8} - \)\(35\!\cdots\!58\)\( p^{36} T^{9} + \)\(22\!\cdots\!58\)\( p^{48} T^{10} - 675735777846 p^{60} T^{11} + p^{72} T^{12} \) | |
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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
−9.430332261412995799504297113276, −9.296944626994536732323096380425, −9.177446929154321283771888924477, −8.361753870197148771102325985956, −7.83950051754511779914798100506, −7.77465540652266369969953427612, −7.07026561414573856416064701330, −6.83245730931968035585536954929, −6.46407882442153452334961908703, −6.29043934589467283317071053813, −6.20105399245093140862734670127, −5.47244395565717974744352970712, −5.36243544676323038304071291283, −5.01729398150977995866673035171, −4.83927921602943592444033528064, −3.75844889258990611288352339447, −3.70298978436624568070546671769, −3.62085323652530120289587286706, −3.44048892740874915779642690196, −2.61641879015446702704345855663, −2.54224139153016316943385007744, −2.20718840248673584202608893524, −1.36819003458998788403911906946, −0.814999744957627247110107349177, −0.35585329146369567201205945431, 0.35585329146369567201205945431, 0.814999744957627247110107349177, 1.36819003458998788403911906946, 2.20718840248673584202608893524, 2.54224139153016316943385007744, 2.61641879015446702704345855663, 3.44048892740874915779642690196, 3.62085323652530120289587286706, 3.70298978436624568070546671769, 3.75844889258990611288352339447, 4.83927921602943592444033528064, 5.01729398150977995866673035171, 5.36243544676323038304071291283, 5.47244395565717974744352970712, 6.20105399245093140862734670127, 6.29043934589467283317071053813, 6.46407882442153452334961908703, 6.83245730931968035585536954929, 7.07026561414573856416064701330, 7.77465540652266369969953427612, 7.83950051754511779914798100506, 8.361753870197148771102325985956, 9.177446929154321283771888924477, 9.296944626994536732323096380425, 9.430332261412995799504297113276
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.