Dirichlet series
| L(s) = 1 | + 48·2-s + 52·3-s + 1.34e3·4-s + 121·5-s + 2.49e3·6-s + 2.05e3·7-s + 2.86e4·8-s − 567·9-s + 5.80e3·10-s + 3.85e3·11-s + 6.98e4·12-s − 1.31e4·13-s + 9.87e4·14-s + 6.29e3·15-s + 5.16e5·16-s + 3.06e4·17-s − 2.72e4·18-s + 4.70e4·19-s + 1.62e5·20-s + 1.07e5·21-s + 1.85e5·22-s − 3.68e4·23-s + 1.49e6·24-s − 2.39e5·25-s − 6.32e5·26-s − 8.63e4·27-s + 2.76e6·28-s + ⋯ |
| L(s) = 1 | + 4.24·2-s + 1.11·3-s + 21/2·4-s + 0.432·5-s + 4.71·6-s + 2.26·7-s + 19.7·8-s − 0.259·9-s + 1.83·10-s + 0.873·11-s + 11.6·12-s − 1.66·13-s + 9.62·14-s + 0.481·15-s + 63/2·16-s + 1.51·17-s − 1.09·18-s + 1.57·19-s + 4.54·20-s + 2.52·21-s + 3.70·22-s − 0.631·23-s + 22.0·24-s − 3.07·25-s − 7.06·26-s − 0.844·27-s + 23.8·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 7^{6} \cdot 13^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.37729\times 10^{10}\) |
| Root analytic conductor: | \(7.54016\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{6} \cdot 7^{6} \cdot 13^{6} ,\ ( \ : [7/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(1809.100303\) |
| \(L(\frac12)\) | \(\approx\) | \(1809.100303\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p^{3} T )^{6} \) |
| 7 | \( ( 1 - p^{3} T )^{6} \) | |
| 13 | \( ( 1 + p^{3} T )^{6} \) | |
| good | 3 | \( 1 - 52 T + 3271 T^{2} - 37744 p T^{3} + 1479991 p T^{4} - 23846924 p^{2} T^{5} + 209991490 p^{3} T^{6} - 23846924 p^{9} T^{7} + 1479991 p^{15} T^{8} - 37744 p^{22} T^{9} + 3271 p^{28} T^{10} - 52 p^{35} T^{11} + p^{42} T^{12} \) |
| 5 | \( 1 - 121 T + 254637 T^{2} - 827074 p T^{3} + 201722777 p^{3} T^{4} + 18414118771 p^{3} T^{5} + 2873408792258 p^{4} T^{6} + 18414118771 p^{10} T^{7} + 201722777 p^{17} T^{8} - 827074 p^{22} T^{9} + 254637 p^{28} T^{10} - 121 p^{35} T^{11} + p^{42} T^{12} \) | |
| 11 | \( 1 - 3856 T + 49203801 T^{2} - 231576988802 T^{3} + 1578240178512985 T^{4} - 7290232256802313462 T^{5} + \)\(33\!\cdots\!50\)\( T^{6} - 7290232256802313462 p^{7} T^{7} + 1578240178512985 p^{14} T^{8} - 231576988802 p^{21} T^{9} + 49203801 p^{28} T^{10} - 3856 p^{35} T^{11} + p^{42} T^{12} \) | |
| 17 | \( 1 - 30606 T + 2445990220 T^{2} - 55884891494802 T^{3} + 2450583176701421223 T^{4} - \)\(42\!\cdots\!88\)\( T^{5} + \)\(13\!\cdots\!32\)\( T^{6} - \)\(42\!\cdots\!88\)\( p^{7} T^{7} + 2450583176701421223 p^{14} T^{8} - 55884891494802 p^{21} T^{9} + 2445990220 p^{28} T^{10} - 30606 p^{35} T^{11} + p^{42} T^{12} \) | |
| 19 | \( 1 - 47087 T + 3061966203 T^{2} - 91620523322844 T^{3} + 3501785677141929943 T^{4} - \)\(74\!\cdots\!45\)\( T^{5} + \)\(28\!\cdots\!62\)\( T^{6} - \)\(74\!\cdots\!45\)\( p^{7} T^{7} + 3501785677141929943 p^{14} T^{8} - 91620523322844 p^{21} T^{9} + 3061966203 p^{28} T^{10} - 47087 p^{35} T^{11} + p^{42} T^{12} \) | |
| 23 | \( 1 + 36839 T + 621057308 p T^{2} + 439512711416747 T^{3} + 95623848116355978212 T^{4} + \)\(23\!\cdots\!87\)\( T^{5} + \)\(39\!\cdots\!94\)\( T^{6} + \)\(23\!\cdots\!87\)\( p^{7} T^{7} + 95623848116355978212 p^{14} T^{8} + 439512711416747 p^{21} T^{9} + 621057308 p^{29} T^{10} + 36839 p^{35} T^{11} + p^{42} T^{12} \) | |
| 29 | \( 1 + 361897 T + 67231243231 T^{2} + 11672355291247876 T^{3} + \)\(21\!\cdots\!25\)\( T^{4} + \)\(31\!\cdots\!83\)\( T^{5} + \)\(39\!\cdots\!62\)\( T^{6} + \)\(31\!\cdots\!83\)\( p^{7} T^{7} + \)\(21\!\cdots\!25\)\( p^{14} T^{8} + 11672355291247876 p^{21} T^{9} + 67231243231 p^{28} T^{10} + 361897 p^{35} T^{11} + p^{42} T^{12} \) | |
| 31 | \( 1 - 354473 T + 156367603310 T^{2} - 41247880024865067 T^{3} + \)\(10\!\cdots\!62\)\( T^{4} - \)\(20\!\cdots\!69\)\( T^{5} + \)\(39\!\cdots\!46\)\( T^{6} - \)\(20\!\cdots\!69\)\( p^{7} T^{7} + \)\(10\!\cdots\!62\)\( p^{14} T^{8} - 41247880024865067 p^{21} T^{9} + 156367603310 p^{28} T^{10} - 354473 p^{35} T^{11} + p^{42} T^{12} \) | |
| 37 | \( 1 - 22362 p T + 501345155991 T^{2} - 241455063821945972 T^{3} + \)\(96\!\cdots\!65\)\( T^{4} - \)\(34\!\cdots\!14\)\( T^{5} + \)\(11\!\cdots\!66\)\( T^{6} - \)\(34\!\cdots\!14\)\( p^{7} T^{7} + \)\(96\!\cdots\!65\)\( p^{14} T^{8} - 241455063821945972 p^{21} T^{9} + 501345155991 p^{28} T^{10} - 22362 p^{36} T^{11} + p^{42} T^{12} \) | |
| 41 | \( 1 - 1093876 T + 1237694784313 T^{2} - 846077623734336748 T^{3} + \)\(57\!\cdots\!47\)\( T^{4} - \)\(29\!\cdots\!96\)\( T^{5} + \)\(14\!\cdots\!78\)\( T^{6} - \)\(29\!\cdots\!96\)\( p^{7} T^{7} + \)\(57\!\cdots\!47\)\( p^{14} T^{8} - 846077623734336748 p^{21} T^{9} + 1237694784313 p^{28} T^{10} - 1093876 p^{35} T^{11} + p^{42} T^{12} \) | |
| 43 | \( 1 - 543449 T + 1110186118117 T^{2} - 721925277832708198 T^{3} + \)\(58\!\cdots\!35\)\( T^{4} - \)\(38\!\cdots\!57\)\( T^{5} + \)\(19\!\cdots\!50\)\( T^{6} - \)\(38\!\cdots\!57\)\( p^{7} T^{7} + \)\(58\!\cdots\!35\)\( p^{14} T^{8} - 721925277832708198 p^{21} T^{9} + 1110186118117 p^{28} T^{10} - 543449 p^{35} T^{11} + p^{42} T^{12} \) | |
| 47 | \( 1 - 1285691 T + 1330248618678 T^{2} - 1671503775332436625 T^{3} + \)\(32\!\cdots\!98\)\( p T^{4} - \)\(12\!\cdots\!39\)\( T^{5} + \)\(94\!\cdots\!74\)\( T^{6} - \)\(12\!\cdots\!39\)\( p^{7} T^{7} + \)\(32\!\cdots\!98\)\( p^{15} T^{8} - 1671503775332436625 p^{21} T^{9} + 1330248618678 p^{28} T^{10} - 1285691 p^{35} T^{11} + p^{42} T^{12} \) | |
| 53 | \( 1 - 263035 T + 3876967963631 T^{2} - 1361233031938827680 T^{3} + \)\(87\!\cdots\!81\)\( T^{4} - \)\(26\!\cdots\!25\)\( T^{5} + \)\(12\!\cdots\!22\)\( T^{6} - \)\(26\!\cdots\!25\)\( p^{7} T^{7} + \)\(87\!\cdots\!81\)\( p^{14} T^{8} - 1361233031938827680 p^{21} T^{9} + 3876967963631 p^{28} T^{10} - 263035 p^{35} T^{11} + p^{42} T^{12} \) | |
| 59 | \( 1 - 6875888 T + 24143420733546 T^{2} - 55395200525090299024 T^{3} + \)\(96\!\cdots\!75\)\( T^{4} - \)\(14\!\cdots\!92\)\( T^{5} + \)\(21\!\cdots\!96\)\( T^{6} - \)\(14\!\cdots\!92\)\( p^{7} T^{7} + \)\(96\!\cdots\!75\)\( p^{14} T^{8} - 55395200525090299024 p^{21} T^{9} + 24143420733546 p^{28} T^{10} - 6875888 p^{35} T^{11} + p^{42} T^{12} \) | |
| 61 | \( 1 - 3602322 T + 19452455758483 T^{2} - 52318199730800783622 T^{3} + \)\(15\!\cdots\!99\)\( T^{4} - \)\(31\!\cdots\!80\)\( T^{5} + \)\(65\!\cdots\!82\)\( T^{6} - \)\(31\!\cdots\!80\)\( p^{7} T^{7} + \)\(15\!\cdots\!99\)\( p^{14} T^{8} - 52318199730800783622 p^{21} T^{9} + 19452455758483 p^{28} T^{10} - 3602322 p^{35} T^{11} + p^{42} T^{12} \) | |
| 67 | \( 1 + 61664 T + 20277941684423 T^{2} - 17947742596857133338 T^{3} + \)\(19\!\cdots\!87\)\( T^{4} - \)\(29\!\cdots\!18\)\( T^{5} + \)\(12\!\cdots\!34\)\( T^{6} - \)\(29\!\cdots\!18\)\( p^{7} T^{7} + \)\(19\!\cdots\!87\)\( p^{14} T^{8} - 17947742596857133338 p^{21} T^{9} + 20277941684423 p^{28} T^{10} + 61664 p^{35} T^{11} + p^{42} T^{12} \) | |
| 71 | \( 1 - 7504176 T + 66679443324852 T^{2} - \)\(32\!\cdots\!56\)\( T^{3} + \)\(16\!\cdots\!03\)\( T^{4} - \)\(57\!\cdots\!64\)\( T^{5} + \)\(19\!\cdots\!40\)\( T^{6} - \)\(57\!\cdots\!64\)\( p^{7} T^{7} + \)\(16\!\cdots\!03\)\( p^{14} T^{8} - \)\(32\!\cdots\!56\)\( p^{21} T^{9} + 66679443324852 p^{28} T^{10} - 7504176 p^{35} T^{11} + p^{42} T^{12} \) | |
| 73 | \( 1 - 2127831 T + 32357792542600 T^{2} - 60212023712742224895 T^{3} + \)\(66\!\cdots\!82\)\( T^{4} - \)\(10\!\cdots\!39\)\( T^{5} + \)\(85\!\cdots\!80\)\( T^{6} - \)\(10\!\cdots\!39\)\( p^{7} T^{7} + \)\(66\!\cdots\!82\)\( p^{14} T^{8} - 60212023712742224895 p^{21} T^{9} + 32357792542600 p^{28} T^{10} - 2127831 p^{35} T^{11} + p^{42} T^{12} \) | |
| 79 | \( 1 + 91555 T + 84233147232176 T^{2} - 12596216102211078885 T^{3} + \)\(32\!\cdots\!24\)\( T^{4} - \)\(90\!\cdots\!25\)\( T^{5} + \)\(77\!\cdots\!94\)\( T^{6} - \)\(90\!\cdots\!25\)\( p^{7} T^{7} + \)\(32\!\cdots\!24\)\( p^{14} T^{8} - 12596216102211078885 p^{21} T^{9} + 84233147232176 p^{28} T^{10} + 91555 p^{35} T^{11} + p^{42} T^{12} \) | |
| 83 | \( 1 - 2372685 T + 69345088852495 T^{2} - \)\(24\!\cdots\!24\)\( T^{3} + \)\(32\!\cdots\!19\)\( T^{4} - \)\(92\!\cdots\!43\)\( T^{5} + \)\(10\!\cdots\!50\)\( T^{6} - \)\(92\!\cdots\!43\)\( p^{7} T^{7} + \)\(32\!\cdots\!19\)\( p^{14} T^{8} - \)\(24\!\cdots\!24\)\( p^{21} T^{9} + 69345088852495 p^{28} T^{10} - 2372685 p^{35} T^{11} + p^{42} T^{12} \) | |
| 89 | \( 1 - 12955563 T + 244395325512325 T^{2} - \)\(23\!\cdots\!02\)\( T^{3} + \)\(26\!\cdots\!93\)\( T^{4} - \)\(19\!\cdots\!91\)\( T^{5} + \)\(15\!\cdots\!50\)\( T^{6} - \)\(19\!\cdots\!91\)\( p^{7} T^{7} + \)\(26\!\cdots\!93\)\( p^{14} T^{8} - \)\(23\!\cdots\!02\)\( p^{21} T^{9} + 244395325512325 p^{28} T^{10} - 12955563 p^{35} T^{11} + p^{42} T^{12} \) | |
| 97 | \( 1 - 6167451 T + 247551377694136 T^{2} - \)\(16\!\cdots\!11\)\( T^{3} + \)\(37\!\cdots\!14\)\( T^{4} - \)\(23\!\cdots\!27\)\( T^{5} + \)\(35\!\cdots\!32\)\( T^{6} - \)\(23\!\cdots\!27\)\( p^{7} T^{7} + \)\(37\!\cdots\!14\)\( p^{14} T^{8} - \)\(16\!\cdots\!11\)\( p^{21} T^{9} + 247551377694136 p^{28} T^{10} - 6167451 p^{35} T^{11} + p^{42} T^{12} \) | |
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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
−5.69730183717794529580567265171, −5.21647747923201798065859710425, −5.13699248837923956242039371766, −5.02587439245679178700048724944, −4.78487306332019921162229015487, −4.71059777927623750282426335433, −4.27351368644735083037385276936, −4.11251520600355360819038306910, −3.88925893801486566075744711178, −3.80290852840397713565171616099, −3.72122976920686279667807392937, −3.57692597101051177660070036955, −3.31659259412679806759755096044, −2.68223761641362730445391457697, −2.60734552775945424196164116358, −2.49877004975501927685602817428, −2.26781511233192656949616039918, −2.22615756762275061223509841324, −2.18093135557694523528667474566, −1.70931481254472736448362039874, −1.45607716495239178272996600550, −1.10921746626363360465203127598, −0.864945111191744958991543793801, −0.63270519955703317367709165747, −0.62447272356984145661782293638, 0.62447272356984145661782293638, 0.63270519955703317367709165747, 0.864945111191744958991543793801, 1.10921746626363360465203127598, 1.45607716495239178272996600550, 1.70931481254472736448362039874, 2.18093135557694523528667474566, 2.22615756762275061223509841324, 2.26781511233192656949616039918, 2.49877004975501927685602817428, 2.60734552775945424196164116358, 2.68223761641362730445391457697, 3.31659259412679806759755096044, 3.57692597101051177660070036955, 3.72122976920686279667807392937, 3.80290852840397713565171616099, 3.88925893801486566075744711178, 4.11251520600355360819038306910, 4.27351368644735083037385276936, 4.71059777927623750282426335433, 4.78487306332019921162229015487, 5.02587439245679178700048724944, 5.13699248837923956242039371766, 5.21647747923201798065859710425, 5.69730183717794529580567265171
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.