Dirichlet series
| L(s) = 1 | − 12·2-s + 9·3-s + 48·4-s − 54·5-s − 108·6-s − 132·7-s + 128·8-s − 48·9-s + 648·10-s − 315·11-s + 432·12-s − 744·13-s + 1.58e3·14-s − 486·15-s − 2.30e3·16-s + 2.89e3·17-s + 576·18-s + 2.26e3·19-s − 2.59e3·20-s − 1.18e3·21-s + 3.78e3·22-s − 3.16e3·23-s + 1.15e3·24-s + 4.70e3·25-s + 8.92e3·26-s + 5.37e3·27-s − 6.33e3·28-s + ⋯ |
| L(s) = 1 | − 2.12·2-s + 0.577·3-s + 3/2·4-s − 0.965·5-s − 1.22·6-s − 1.01·7-s + 0.707·8-s − 0.197·9-s + 2.04·10-s − 0.784·11-s + 0.866·12-s − 1.22·13-s + 2.15·14-s − 0.557·15-s − 9/4·16-s + 2.43·17-s + 0.419·18-s + 1.43·19-s − 1.44·20-s − 0.587·21-s + 1.66·22-s − 1.24·23-s + 0.408·24-s + 1.50·25-s + 2.59·26-s + 1.41·27-s − 1.52·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut & 34012224 ^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 34012224 ^{s/2} \, \Gamma_{\C}(s+5/2)^{6} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(34012224\) = \(2^{6} \cdot 3^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(578.893\) |
| Root analytic conductor: | \(1.69909\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 34012224,\ (\ :[5/2]^{6}),\ 1)\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(0.2549047807\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2549047807\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{2} T + p^{4} T^{2} )^{3} \) |
| 3 | \( 1 - p^{2} T + 43 p T^{2} - 86 p^{4} T^{3} + 43 p^{6} T^{4} - p^{12} T^{5} + p^{15} T^{6} \) | |
| good | 5 | \( 1 + 54 T - 1788 T^{2} + 10584 T^{3} + 854952 T^{4} - 487704186 T^{5} - 21775421594 T^{6} - 487704186 p^{5} T^{7} + 854952 p^{10} T^{8} + 10584 p^{15} T^{9} - 1788 p^{20} T^{10} + 54 p^{25} T^{11} + p^{30} T^{12} \) |
| 7 | \( 1 + 132 T + 9930 T^{2} + 2383268 T^{3} + 22216338 T^{4} - 1188047664 T^{5} + 4115243377158 T^{6} - 1188047664 p^{5} T^{7} + 22216338 p^{10} T^{8} + 2383268 p^{15} T^{9} + 9930 p^{20} T^{10} + 132 p^{25} T^{11} + p^{30} T^{12} \) | |
| 11 | \( 1 + 315 T - 222063 T^{2} - 18182214 T^{3} + 32678625651 T^{4} - 477210982239 p T^{5} - 7280422055915978 T^{6} - 477210982239 p^{6} T^{7} + 32678625651 p^{10} T^{8} - 18182214 p^{15} T^{9} - 222063 p^{20} T^{10} + 315 p^{25} T^{11} + p^{30} T^{12} \) | |
| 13 | \( 1 + 744 T + 110316 T^{2} - 5563444 T^{3} - 92859229188 T^{4} - 67995162383184 T^{5} - 18766393745055906 T^{6} - 67995162383184 p^{5} T^{7} - 92859229188 p^{10} T^{8} - 5563444 p^{15} T^{9} + 110316 p^{20} T^{10} + 744 p^{25} T^{11} + p^{30} T^{12} \) | |
| 17 | \( ( 1 - 1449 T + 3759531 T^{2} - 3184553142 T^{3} + 3759531 p^{5} T^{4} - 1449 p^{10} T^{5} + p^{15} T^{6} )^{2} \) | |
| 19 | \( ( 1 - 1131 T + 5928369 T^{2} - 5953391858 T^{3} + 5928369 p^{5} T^{4} - 1131 p^{10} T^{5} + p^{15} T^{6} )^{2} \) | |
| 23 | \( 1 + 3168 T - 11449086 T^{2} - 14347077444 T^{3} + 192790258568754 T^{4} + 140383346504289372 T^{5} - \)\(11\!\cdots\!18\)\( T^{6} + 140383346504289372 p^{5} T^{7} + 192790258568754 p^{10} T^{8} - 14347077444 p^{15} T^{9} - 11449086 p^{20} T^{10} + 3168 p^{25} T^{11} + p^{30} T^{12} \) | |
| 29 | \( 1 + 5148 T - 22104564 T^{2} - 185150934252 T^{3} + 244953744801108 T^{4} + 2482021182847364820 T^{5} + \)\(50\!\cdots\!82\)\( T^{6} + 2482021182847364820 p^{5} T^{7} + 244953744801108 p^{10} T^{8} - 185150934252 p^{15} T^{9} - 22104564 p^{20} T^{10} + 5148 p^{25} T^{11} + p^{30} T^{12} \) | |
| 31 | \( 1 + 8610 T - 10689342 T^{2} - 137042221612 T^{3} + 1032348577619862 T^{4} + 2025683261171583642 T^{5} - \)\(31\!\cdots\!06\)\( T^{6} + 2025683261171583642 p^{5} T^{7} + 1032348577619862 p^{10} T^{8} - 137042221612 p^{15} T^{9} - 10689342 p^{20} T^{10} + 8610 p^{25} T^{11} + p^{30} T^{12} \) | |
| 37 | \( ( 1 - 19968 T + 327446979 T^{2} - 2957339330768 T^{3} + 327446979 p^{5} T^{4} - 19968 p^{10} T^{5} + p^{15} T^{6} )^{2} \) | |
| 41 | \( 1 - 5049 T - 49911369 T^{2} - 2355608603160 T^{3} + 1935356544736665 T^{4} + 94902326907305939361 T^{5} + \)\(30\!\cdots\!26\)\( T^{6} + 94902326907305939361 p^{5} T^{7} + 1935356544736665 p^{10} T^{8} - 2355608603160 p^{15} T^{9} - 49911369 p^{20} T^{10} - 5049 p^{25} T^{11} + p^{30} T^{12} \) | |
| 43 | \( 1 + 31389 T + 371875485 T^{2} + 42403467926 p T^{3} + 12515413892662971 T^{4} + \)\(36\!\cdots\!65\)\( T^{5} + \)\(57\!\cdots\!06\)\( T^{6} + \)\(36\!\cdots\!65\)\( p^{5} T^{7} + 12515413892662971 p^{10} T^{8} + 42403467926 p^{16} T^{9} + 371875485 p^{20} T^{10} + 31389 p^{25} T^{11} + p^{30} T^{12} \) | |
| 47 | \( 1 - 12924 T - 448629342 T^{2} + 4068633746340 T^{3} + 160129591294416234 T^{4} - \)\(73\!\cdots\!76\)\( T^{5} - \)\(35\!\cdots\!10\)\( T^{6} - \)\(73\!\cdots\!76\)\( p^{5} T^{7} + 160129591294416234 p^{10} T^{8} + 4068633746340 p^{15} T^{9} - 448629342 p^{20} T^{10} - 12924 p^{25} T^{11} + p^{30} T^{12} \) | |
| 53 | \( ( 1 + 48024 T + 1811738067 T^{2} + 41931353529216 T^{3} + 1811738067 p^{5} T^{4} + 48024 p^{10} T^{5} + p^{15} T^{6} )^{2} \) | |
| 59 | \( 1 - 62955 T + 1128633393 T^{2} - 9982647295866 T^{3} + 705724247444789571 T^{4} - \)\(15\!\cdots\!79\)\( T^{5} + \)\(56\!\cdots\!06\)\( T^{6} - \)\(15\!\cdots\!79\)\( p^{5} T^{7} + 705724247444789571 p^{10} T^{8} - 9982647295866 p^{15} T^{9} + 1128633393 p^{20} T^{10} - 62955 p^{25} T^{11} + p^{30} T^{12} \) | |
| 61 | \( 1 + 75966 T + 1651965492 T^{2} + 45532183560440 T^{3} + 3331090328104364760 T^{4} + \)\(76\!\cdots\!06\)\( T^{5} + \)\(63\!\cdots\!14\)\( T^{6} + \)\(76\!\cdots\!06\)\( p^{5} T^{7} + 3331090328104364760 p^{10} T^{8} + 45532183560440 p^{15} T^{9} + 1651965492 p^{20} T^{10} + 75966 p^{25} T^{11} + p^{30} T^{12} \) | |
| 67 | \( 1 + 32991 T - 1801789179 T^{2} - 76747890991738 T^{3} + 1612255572283172571 T^{4} + \)\(51\!\cdots\!59\)\( T^{5} - \)\(93\!\cdots\!82\)\( T^{6} + \)\(51\!\cdots\!59\)\( p^{5} T^{7} + 1612255572283172571 p^{10} T^{8} - 76747890991738 p^{15} T^{9} - 1801789179 p^{20} T^{10} + 32991 p^{25} T^{11} + p^{30} T^{12} \) | |
| 71 | \( ( 1 + 64836 T + 5330360517 T^{2} + 233818077065976 T^{3} + 5330360517 p^{5} T^{4} + 64836 p^{10} T^{5} + p^{15} T^{6} )^{2} \) | |
| 73 | \( ( 1 + 4233 T + 1481704827 T^{2} + 31872982860070 T^{3} + 1481704827 p^{5} T^{4} + 4233 p^{10} T^{5} + p^{15} T^{6} )^{2} \) | |
| 79 | \( 1 - 89202 T - 946471758 T^{2} + 73104980305676 T^{3} + 13192286849737226406 T^{4} + \)\(18\!\cdots\!50\)\( T^{5} - \)\(72\!\cdots\!62\)\( T^{6} + \)\(18\!\cdots\!50\)\( p^{5} T^{7} + 13192286849737226406 p^{10} T^{8} + 73104980305676 p^{15} T^{9} - 946471758 p^{20} T^{10} - 89202 p^{25} T^{11} + p^{30} T^{12} \) | |
| 83 | \( 1 - 32634 T - 7761884658 T^{2} + 19287559006164 T^{3} + 38129210867128845714 T^{4} + \)\(40\!\cdots\!90\)\( T^{5} - \)\(17\!\cdots\!58\)\( T^{6} + \)\(40\!\cdots\!90\)\( p^{5} T^{7} + 38129210867128845714 p^{10} T^{8} + 19287559006164 p^{15} T^{9} - 7761884658 p^{20} T^{10} - 32634 p^{25} T^{11} + p^{30} T^{12} \) | |
| 89 | \( ( 1 - 33066 T + 16131261399 T^{2} - 360180435327660 T^{3} + 16131261399 p^{5} T^{4} - 33066 p^{10} T^{5} + p^{15} T^{6} )^{2} \) | |
| 97 | \( 1 - 46245 T - 17067186177 T^{2} + 116084978368376 T^{3} + \)\(18\!\cdots\!05\)\( T^{4} + \)\(22\!\cdots\!89\)\( T^{5} - \)\(18\!\cdots\!78\)\( T^{6} + \)\(22\!\cdots\!89\)\( p^{5} T^{7} + \)\(18\!\cdots\!05\)\( p^{10} T^{8} + 116084978368376 p^{15} T^{9} - 17067186177 p^{20} T^{10} - 46245 p^{25} T^{11} + p^{30} T^{12} \) | |
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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
−9.836308933444155697069781855300, −9.616811875477401714869018676737, −9.475077200999163593879622889209, −9.281135919991846334575328180449, −9.191918292527484390043958788067, −8.669307454575246483405434428531, −8.209909422166116065329542666778, −8.016668984762849669723093966124, −7.82908183812986805363425832654, −7.73228362083177067645477637331, −7.35859196614345558665499002618, −7.33103831796893653089430074620, −6.66979716421108013218873373971, −6.13191726385951281914000296910, −6.00974972257582840385169347036, −5.34074953897671859287786828179, −5.04791351640243052345209877504, −4.56636687833116194394944606844, −4.20451398805836135541476945585, −3.35908639154472099665748651899, −3.06793779425881685658265795884, −2.87567830729649205634322786998, −1.71573879529527477054922217364, −0.916430588039136972737422307563, −0.31637154417658173846555528120, 0.31637154417658173846555528120, 0.916430588039136972737422307563, 1.71573879529527477054922217364, 2.87567830729649205634322786998, 3.06793779425881685658265795884, 3.35908639154472099665748651899, 4.20451398805836135541476945585, 4.56636687833116194394944606844, 5.04791351640243052345209877504, 5.34074953897671859287786828179, 6.00974972257582840385169347036, 6.13191726385951281914000296910, 6.66979716421108013218873373971, 7.33103831796893653089430074620, 7.35859196614345558665499002618, 7.73228362083177067645477637331, 7.82908183812986805363425832654, 8.016668984762849669723093966124, 8.209909422166116065329542666778, 8.669307454575246483405434428531, 9.191918292527484390043958788067, 9.281135919991846334575328180449, 9.475077200999163593879622889209, 9.616811875477401714869018676737, 9.836308933444155697069781855300
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.