Dirichlet series
| L(s) = 1 | − 48·2-s − 162·3-s + 1.34e3·4-s − 181·5-s + 7.77e3·6-s + 2.05e3·7-s − 2.86e4·8-s + 1.53e4·9-s + 8.68e3·10-s − 6.13e3·11-s − 2.17e5·12-s + 1.31e4·13-s − 9.87e4·14-s + 2.93e4·15-s + 5.16e5·16-s − 3.46e4·17-s − 7.34e5·18-s − 4.08e3·19-s − 2.43e5·20-s − 3.33e5·21-s + 2.94e5·22-s + 1.51e3·23-s + 4.64e6·24-s − 1.39e5·25-s − 6.32e5·26-s − 1.10e6·27-s + 2.76e6·28-s + ⋯ |
| L(s) = 1 | − 4.24·2-s − 3.46·3-s + 21/2·4-s − 0.647·5-s + 14.6·6-s + 2.26·7-s − 19.7·8-s + 7·9-s + 2.74·10-s − 1.38·11-s − 36.3·12-s + 1.66·13-s − 9.62·14-s + 2.24·15-s + 63/2·16-s − 1.70·17-s − 29.6·18-s − 0.136·19-s − 6.79·20-s − 7.85·21-s + 5.89·22-s + 0.0259·23-s + 68.5·24-s − 1.78·25-s − 7.06·26-s − 10.7·27-s + 23.8·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{6}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{6}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.46205\times 10^{13}\) |
| Root analytic conductor: | \(13.0599\) |
| Motivic weight: | \(7\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 2^{6} \cdot 3^{6} \cdot 7^{6} \cdot 13^{6} ,\ ( \ : [7/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(4)\) | \(\approx\) | \(0.01377467853\) |
| \(L(\frac12)\) | \(\approx\) | \(0.01377467853\) |
| \(L(\frac{9}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{3} T )^{6} \) |
| 3 | \( ( 1 + p^{3} T )^{6} \) | |
| 7 | \( ( 1 - p^{3} T )^{6} \) | |
| 13 | \( ( 1 - p^{3} T )^{6} \) | |
| good | 5 | \( 1 + 181 T + 172451 T^{2} + 48606676 T^{3} + 4765770017 p T^{4} + 221007332071 p^{2} T^{5} + 17846180438678 p^{3} T^{6} + 221007332071 p^{9} T^{7} + 4765770017 p^{15} T^{8} + 48606676 p^{21} T^{9} + 172451 p^{28} T^{10} + 181 p^{35} T^{11} + p^{42} T^{12} \) |
| 11 | \( 1 + 6130 T + 73878881 T^{2} + 363582119260 T^{3} + 2872263961628131 T^{4} + 11352352289566105714 T^{5} + \)\(67\!\cdots\!46\)\( T^{6} + 11352352289566105714 p^{7} T^{7} + 2872263961628131 p^{14} T^{8} + 363582119260 p^{21} T^{9} + 73878881 p^{28} T^{10} + 6130 p^{35} T^{11} + p^{42} T^{12} \) | |
| 17 | \( 1 + 34610 T + 1632892913 T^{2} + 41810137496408 T^{3} + 1158910817174754007 T^{4} + \)\(24\!\cdots\!70\)\( T^{5} + \)\(54\!\cdots\!10\)\( T^{6} + \)\(24\!\cdots\!70\)\( p^{7} T^{7} + 1158910817174754007 p^{14} T^{8} + 41810137496408 p^{21} T^{9} + 1632892913 p^{28} T^{10} + 34610 p^{35} T^{11} + p^{42} T^{12} \) | |
| 19 | \( 1 + 215 p T + 628489437 T^{2} + 8764452098062 T^{3} + 399360761818646399 T^{4} + \)\(15\!\cdots\!17\)\( T^{5} + \)\(75\!\cdots\!66\)\( T^{6} + \)\(15\!\cdots\!17\)\( p^{7} T^{7} + 399360761818646399 p^{14} T^{8} + 8764452098062 p^{21} T^{9} + 628489437 p^{28} T^{10} + 215 p^{36} T^{11} + p^{42} T^{12} \) | |
| 23 | \( 1 - 1515 T - 92298119 T^{2} - 21064018465266 T^{3} + 13804498961212728671 T^{4} + \)\(18\!\cdots\!41\)\( T^{5} + \)\(80\!\cdots\!62\)\( T^{6} + \)\(18\!\cdots\!41\)\( p^{7} T^{7} + 13804498961212728671 p^{14} T^{8} - 21064018465266 p^{21} T^{9} - 92298119 p^{28} T^{10} - 1515 p^{35} T^{11} + p^{42} T^{12} \) | |
| 29 | \( 1 + 59395 T + 39789149471 T^{2} + 4906546112764132 T^{3} + \)\(98\!\cdots\!25\)\( T^{4} + \)\(15\!\cdots\!37\)\( T^{5} + \)\(18\!\cdots\!90\)\( T^{6} + \)\(15\!\cdots\!37\)\( p^{7} T^{7} + \)\(98\!\cdots\!25\)\( p^{14} T^{8} + 4906546112764132 p^{21} T^{9} + 39789149471 p^{28} T^{10} + 59395 p^{35} T^{11} + p^{42} T^{12} \) | |
| 31 | \( 1 - 478241 T + 237860670126 T^{2} - 2214380936420517 p T^{3} + \)\(19\!\cdots\!15\)\( T^{4} - \)\(38\!\cdots\!90\)\( T^{5} + \)\(73\!\cdots\!40\)\( T^{6} - \)\(38\!\cdots\!90\)\( p^{7} T^{7} + \)\(19\!\cdots\!15\)\( p^{14} T^{8} - 2214380936420517 p^{22} T^{9} + 237860670126 p^{28} T^{10} - 478241 p^{35} T^{11} + p^{42} T^{12} \) | |
| 37 | \( 1 - 574310 T + 451809039061 T^{2} - 173437719987720524 T^{3} + \)\(87\!\cdots\!59\)\( T^{4} - \)\(27\!\cdots\!30\)\( T^{5} + \)\(10\!\cdots\!66\)\( T^{6} - \)\(27\!\cdots\!30\)\( p^{7} T^{7} + \)\(87\!\cdots\!59\)\( p^{14} T^{8} - 173437719987720524 p^{21} T^{9} + 451809039061 p^{28} T^{10} - 574310 p^{35} T^{11} + p^{42} T^{12} \) | |
| 41 | \( 1 - 201552 T + 545722934854 T^{2} + 49538385650050080 T^{3} + \)\(32\!\cdots\!43\)\( p T^{4} + \)\(33\!\cdots\!80\)\( T^{5} + \)\(28\!\cdots\!08\)\( T^{6} + \)\(33\!\cdots\!80\)\( p^{7} T^{7} + \)\(32\!\cdots\!43\)\( p^{15} T^{8} + 49538385650050080 p^{21} T^{9} + 545722934854 p^{28} T^{10} - 201552 p^{35} T^{11} + p^{42} T^{12} \) | |
| 43 | \( 1 - 728605 T + 1198613589437 T^{2} - 522684249560574558 T^{3} + \)\(57\!\cdots\!91\)\( T^{4} - \)\(18\!\cdots\!73\)\( T^{5} + \)\(17\!\cdots\!50\)\( T^{6} - \)\(18\!\cdots\!73\)\( p^{7} T^{7} + \)\(57\!\cdots\!91\)\( p^{14} T^{8} - 522684249560574558 p^{21} T^{9} + 1198613589437 p^{28} T^{10} - 728605 p^{35} T^{11} + p^{42} T^{12} \) | |
| 47 | \( 1 - 227615 T + 1752314402014 T^{2} - 728490597739277477 T^{3} + \)\(16\!\cdots\!19\)\( T^{4} - \)\(68\!\cdots\!86\)\( T^{5} + \)\(99\!\cdots\!32\)\( T^{6} - \)\(68\!\cdots\!86\)\( p^{7} T^{7} + \)\(16\!\cdots\!19\)\( p^{14} T^{8} - 728490597739277477 p^{21} T^{9} + 1752314402014 p^{28} T^{10} - 227615 p^{35} T^{11} + p^{42} T^{12} \) | |
| 53 | \( 1 - 26321 T + 2804984774212 T^{2} - 1344408602257045961 T^{3} + \)\(21\!\cdots\!83\)\( T^{4} - \)\(43\!\cdots\!58\)\( T^{5} + \)\(44\!\cdots\!32\)\( T^{6} - \)\(43\!\cdots\!58\)\( p^{7} T^{7} + \)\(21\!\cdots\!83\)\( p^{14} T^{8} - 1344408602257045961 p^{21} T^{9} + 2804984774212 p^{28} T^{10} - 26321 p^{35} T^{11} + p^{42} T^{12} \) | |
| 59 | \( 1 - 478280 T + 7630587428306 T^{2} - 7154170786681516664 T^{3} + \)\(31\!\cdots\!91\)\( T^{4} - \)\(34\!\cdots\!56\)\( T^{5} + \)\(90\!\cdots\!44\)\( T^{6} - \)\(34\!\cdots\!56\)\( p^{7} T^{7} + \)\(31\!\cdots\!91\)\( p^{14} T^{8} - 7154170786681516664 p^{21} T^{9} + 7630587428306 p^{28} T^{10} - 478280 p^{35} T^{11} + p^{42} T^{12} \) | |
| 61 | \( 1 + 501406 T + 9573794109895 T^{2} + 5297177154034589530 T^{3} + \)\(49\!\cdots\!03\)\( T^{4} + \)\(33\!\cdots\!32\)\( T^{5} + \)\(18\!\cdots\!38\)\( T^{6} + \)\(33\!\cdots\!32\)\( p^{7} T^{7} + \)\(49\!\cdots\!03\)\( p^{14} T^{8} + 5297177154034589530 p^{21} T^{9} + 9573794109895 p^{28} T^{10} + 501406 p^{35} T^{11} + p^{42} T^{12} \) | |
| 67 | \( 1 + 3156366 T + 26572618390294 T^{2} + 64836786457933537002 T^{3} + \)\(33\!\cdots\!39\)\( T^{4} + \)\(66\!\cdots\!04\)\( T^{5} + \)\(25\!\cdots\!08\)\( T^{6} + \)\(66\!\cdots\!04\)\( p^{7} T^{7} + \)\(33\!\cdots\!39\)\( p^{14} T^{8} + 64836786457933537002 p^{21} T^{9} + 26572618390294 p^{28} T^{10} + 3156366 p^{35} T^{11} + p^{42} T^{12} \) | |
| 71 | \( 1 + 2003644 T + 17449321420378 T^{2} + 70451300679120757300 T^{3} + \)\(27\!\cdots\!55\)\( T^{4} + \)\(75\!\cdots\!76\)\( T^{5} + \)\(33\!\cdots\!96\)\( T^{6} + \)\(75\!\cdots\!76\)\( p^{7} T^{7} + \)\(27\!\cdots\!55\)\( p^{14} T^{8} + 70451300679120757300 p^{21} T^{9} + 17449321420378 p^{28} T^{10} + 2003644 p^{35} T^{11} + p^{42} T^{12} \) | |
| 73 | \( 1 - 3659111 T + 56428660594327 T^{2} - \)\(16\!\cdots\!12\)\( T^{3} + \)\(14\!\cdots\!17\)\( T^{4} - \)\(32\!\cdots\!13\)\( T^{5} + \)\(19\!\cdots\!86\)\( T^{6} - \)\(32\!\cdots\!13\)\( p^{7} T^{7} + \)\(14\!\cdots\!17\)\( p^{14} T^{8} - \)\(16\!\cdots\!12\)\( p^{21} T^{9} + 56428660594327 p^{28} T^{10} - 3659111 p^{35} T^{11} + p^{42} T^{12} \) | |
| 79 | \( 1 - 1131065 T + 68147909010706 T^{2} - 87827526806803965719 T^{3} + \)\(25\!\cdots\!43\)\( T^{4} - \)\(30\!\cdots\!02\)\( T^{5} + \)\(60\!\cdots\!60\)\( T^{6} - \)\(30\!\cdots\!02\)\( p^{7} T^{7} + \)\(25\!\cdots\!43\)\( p^{14} T^{8} - 87827526806803965719 p^{21} T^{9} + 68147909010706 p^{28} T^{10} - 1131065 p^{35} T^{11} + p^{42} T^{12} \) | |
| 83 | \( 1 + 9629297 T + 156192377251238 T^{2} + \)\(10\!\cdots\!87\)\( T^{3} + \)\(99\!\cdots\!91\)\( T^{4} + \)\(50\!\cdots\!90\)\( T^{5} + \)\(35\!\cdots\!04\)\( T^{6} + \)\(50\!\cdots\!90\)\( p^{7} T^{7} + \)\(99\!\cdots\!91\)\( p^{14} T^{8} + \)\(10\!\cdots\!87\)\( p^{21} T^{9} + 156192377251238 p^{28} T^{10} + 9629297 p^{35} T^{11} + p^{42} T^{12} \) | |
| 89 | \( 1 + 21977377 T + 360109613594828 T^{2} + \)\(43\!\cdots\!29\)\( T^{3} + \)\(43\!\cdots\!75\)\( T^{4} + \)\(37\!\cdots\!78\)\( T^{5} + \)\(26\!\cdots\!32\)\( T^{6} + \)\(37\!\cdots\!78\)\( p^{7} T^{7} + \)\(43\!\cdots\!75\)\( p^{14} T^{8} + \)\(43\!\cdots\!29\)\( p^{21} T^{9} + 360109613594828 p^{28} T^{10} + 21977377 p^{35} T^{11} + p^{42} T^{12} \) | |
| 97 | \( 1 + 26386649 T + 593431855669140 T^{2} + \)\(89\!\cdots\!37\)\( T^{3} + \)\(12\!\cdots\!11\)\( T^{4} + \)\(12\!\cdots\!42\)\( T^{5} + \)\(12\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!42\)\( p^{7} T^{7} + \)\(12\!\cdots\!11\)\( p^{14} T^{8} + \)\(89\!\cdots\!37\)\( p^{21} T^{9} + 593431855669140 p^{28} T^{10} + 26386649 p^{35} T^{11} + p^{42} T^{12} \) | |
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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
−4.98505183358950022230732424265, −4.20020026931169665102405277838, −4.12897588011408202744918128171, −4.07238953546663807016145131618, −4.06923479298628867710761035102, −4.02609944542971233022375198941, −4.02320768703189355851589848185, −2.99754209603261415500120563364, −2.95524763766611192832219562136, −2.65767816570691855737131662852, −2.61077719414962670317171931352, −2.57472850056236284882665785411, −2.48540638861174702096703143854, −1.68913868689695143008298927731, −1.66967850830902865010638128329, −1.66636501155329652403034472953, −1.51469493844167602245383963436, −1.48564716418333705211804844895, −1.23902932097156640015693955740, −0.78056801162489954660477447004, −0.69806972361556817591945281779, −0.69390456927513342232626589775, −0.50266973182986301857552390872, −0.27358890956052912298421258971, −0.04530235267975639826289814978, 0.04530235267975639826289814978, 0.27358890956052912298421258971, 0.50266973182986301857552390872, 0.69390456927513342232626589775, 0.69806972361556817591945281779, 0.78056801162489954660477447004, 1.23902932097156640015693955740, 1.48564716418333705211804844895, 1.51469493844167602245383963436, 1.66636501155329652403034472953, 1.66967850830902865010638128329, 1.68913868689695143008298927731, 2.48540638861174702096703143854, 2.57472850056236284882665785411, 2.61077719414962670317171931352, 2.65767816570691855737131662852, 2.95524763766611192832219562136, 2.99754209603261415500120563364, 4.02320768703189355851589848185, 4.02609944542971233022375198941, 4.06923479298628867710761035102, 4.07238953546663807016145131618, 4.12897588011408202744918128171, 4.20020026931169665102405277838, 4.98505183358950022230732424265
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.