Dirichlet series
| L(s) = 1 | + 18·3-s − 24·4-s + 113·9-s − 432·12-s + 336·16-s − 198·17-s − 534·23-s + 382·25-s + 374·27-s − 238·29-s − 2.71e3·36-s − 856·43-s + 6.04e3·48-s + 1.15e3·49-s − 3.56e3·51-s + 178·53-s + 3.40e3·61-s − 3.58e3·64-s + 4.75e3·68-s − 9.61e3·69-s + 6.87e3·75-s − 1.75e3·79-s + 2.89e3·81-s − 4.28e3·87-s + 1.28e4·92-s − 9.16e3·100-s + 5.49e3·101-s + ⋯ |
| L(s) = 1 | + 3.46·3-s − 3·4-s + 4.18·9-s − 10.3·12-s + 21/4·16-s − 2.82·17-s − 4.84·23-s + 3.05·25-s + 2.66·27-s − 1.52·29-s − 12.5·36-s − 3.03·43-s + 18.1·48-s + 3.37·49-s − 9.78·51-s + 0.461·53-s + 7.15·61-s − 7·64-s + 8.47·68-s − 16.7·69-s + 10.5·75-s − 2.49·79-s + 3.96·81-s − 5.27·87-s + 14.5·92-s − 9.16·100-s + 5.41·101-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 13^{24}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 13^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.95724\times 10^{15}\) |
| Root analytic conductor: | \(4.46571\) |
| Motivic weight: | \(3\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 13^{24} ,\ ( \ : [3/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(2)\) | \(\approx\) | \(0.1973098938\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1973098938\) |
| \(L(\frac{5}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{2} T^{2} )^{6} \) |
| 13 | \( 1 \) | |
| good | 3 | \( ( 1 - p^{2} T + 65 T^{2} - 484 T^{3} + 308 p^{2} T^{4} - 14446 T^{5} + 82285 T^{6} - 14446 p^{3} T^{7} + 308 p^{8} T^{8} - 484 p^{9} T^{9} + 65 p^{12} T^{10} - p^{17} T^{11} + p^{18} T^{12} )^{2} \) |
| 5 | \( 1 - 382 T^{2} + 93199 T^{4} - 3539669 p T^{6} + 2754615289 T^{8} - 389109534481 T^{10} + 50512702160206 T^{12} - 389109534481 p^{6} T^{14} + 2754615289 p^{12} T^{16} - 3539669 p^{19} T^{18} + 93199 p^{24} T^{20} - 382 p^{30} T^{22} + p^{36} T^{24} \) | |
| 7 | \( 1 - 1159 T^{2} + 813634 T^{4} - 415626322 T^{6} + 25311558955 p T^{8} - 1406264176663 p^{2} T^{10} + 496947802324576 p^{2} T^{12} - 1406264176663 p^{8} T^{14} + 25311558955 p^{13} T^{16} - 415626322 p^{18} T^{18} + 813634 p^{24} T^{20} - 1159 p^{30} T^{22} + p^{36} T^{24} \) | |
| 11 | \( 1 - 8117 T^{2} + 33989553 T^{4} - 98393356134 T^{6} + 219078606548874 T^{8} - 391842639166411830 T^{10} + \)\(57\!\cdots\!35\)\( T^{12} - 391842639166411830 p^{6} T^{14} + 219078606548874 p^{12} T^{16} - 98393356134 p^{18} T^{18} + 33989553 p^{24} T^{20} - 8117 p^{30} T^{22} + p^{36} T^{24} \) | |
| 17 | \( ( 1 + 99 T + 15235 T^{2} + 1418010 T^{3} + 155831450 T^{4} + 11553597880 T^{5} + 895771727153 T^{6} + 11553597880 p^{3} T^{7} + 155831450 p^{6} T^{8} + 1418010 p^{9} T^{9} + 15235 p^{12} T^{10} + 99 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 19 | \( 1 - 36013 T^{2} + 716146137 T^{4} - 10102929361070 T^{6} + 111338486468995570 T^{8} - \)\(10\!\cdots\!14\)\( T^{10} + \)\(74\!\cdots\!39\)\( T^{12} - \)\(10\!\cdots\!14\)\( p^{6} T^{14} + 111338486468995570 p^{12} T^{16} - 10102929361070 p^{18} T^{18} + 716146137 p^{24} T^{20} - 36013 p^{30} T^{22} + p^{36} T^{24} \) | |
| 23 | \( ( 1 + 267 T + 65293 T^{2} + 11352002 T^{3} + 1904634711 T^{4} + 10605914921 p T^{5} + 29697637919118 T^{6} + 10605914921 p^{4} T^{7} + 1904634711 p^{6} T^{8} + 11352002 p^{9} T^{9} + 65293 p^{12} T^{10} + 267 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 29 | \( ( 1 + 119 T + 124344 T^{2} + 11836678 T^{3} + 6759160701 T^{4} + 516098000019 T^{5} + 7271394875556 p T^{6} + 516098000019 p^{3} T^{7} + 6759160701 p^{6} T^{8} + 11836678 p^{9} T^{9} + 124344 p^{12} T^{10} + 119 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 31 | \( 1 - 241655 T^{2} + 28942573242 T^{4} - 2264770361862706 T^{6} + \)\(12\!\cdots\!57\)\( T^{8} - \)\(55\!\cdots\!07\)\( T^{10} + \)\(18\!\cdots\!20\)\( T^{12} - \)\(55\!\cdots\!07\)\( p^{6} T^{14} + \)\(12\!\cdots\!57\)\( p^{12} T^{16} - 2264770361862706 p^{18} T^{18} + 28942573242 p^{24} T^{20} - 241655 p^{30} T^{22} + p^{36} T^{24} \) | |
| 37 | \( 1 - 419434 T^{2} + 87929952639 T^{4} - 11996835934207205 T^{6} + \)\(11\!\cdots\!33\)\( T^{8} - \)\(87\!\cdots\!61\)\( T^{10} + \)\(50\!\cdots\!86\)\( T^{12} - \)\(87\!\cdots\!61\)\( p^{6} T^{14} + \)\(11\!\cdots\!33\)\( p^{12} T^{16} - 11996835934207205 p^{18} T^{18} + 87929952639 p^{24} T^{20} - 419434 p^{30} T^{22} + p^{36} T^{24} \) | |
| 41 | \( 1 - 435796 T^{2} + 88505255842 T^{4} - 11683055467962385 T^{6} + \)\(12\!\cdots\!47\)\( T^{8} - \)\(10\!\cdots\!94\)\( T^{10} + \)\(79\!\cdots\!91\)\( T^{12} - \)\(10\!\cdots\!94\)\( p^{6} T^{14} + \)\(12\!\cdots\!47\)\( p^{12} T^{16} - 11683055467962385 p^{18} T^{18} + 88505255842 p^{24} T^{20} - 435796 p^{30} T^{22} + p^{36} T^{24} \) | |
| 43 | \( ( 1 + 428 T + 199837 T^{2} + 75716384 T^{3} + 36167871145 T^{4} + 9851740849988 T^{5} + 2980287990273901 T^{6} + 9851740849988 p^{3} T^{7} + 36167871145 p^{6} T^{8} + 75716384 p^{9} T^{9} + 199837 p^{12} T^{10} + 428 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 47 | \( 1 - 453138 T^{2} + 135321770311 T^{4} - 29047321613511285 T^{6} + \)\(49\!\cdots\!41\)\( T^{8} - \)\(67\!\cdots\!21\)\( T^{10} + \)\(76\!\cdots\!02\)\( T^{12} - \)\(67\!\cdots\!21\)\( p^{6} T^{14} + \)\(49\!\cdots\!41\)\( p^{12} T^{16} - 29047321613511285 p^{18} T^{18} + 135321770311 p^{24} T^{20} - 453138 p^{30} T^{22} + p^{36} T^{24} \) | |
| 53 | \( ( 1 - 89 T + 149636 T^{2} + 19578912 T^{3} + 11883866543 T^{4} - 8498293720903 T^{5} + 5004078025030288 T^{6} - 8498293720903 p^{3} T^{7} + 11883866543 p^{6} T^{8} + 19578912 p^{9} T^{9} + 149636 p^{12} T^{10} - 89 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 59 | \( 1 - 526184 T^{2} + 141586970478 T^{4} - 41662826122817849 T^{6} + \)\(13\!\cdots\!71\)\( T^{8} - \)\(29\!\cdots\!54\)\( T^{10} + \)\(57\!\cdots\!15\)\( T^{12} - \)\(29\!\cdots\!54\)\( p^{6} T^{14} + \)\(13\!\cdots\!71\)\( p^{12} T^{16} - 41662826122817849 p^{18} T^{18} + 141586970478 p^{24} T^{20} - 526184 p^{30} T^{22} + p^{36} T^{24} \) | |
| 61 | \( ( 1 - 1704 T + 1938703 T^{2} - 1635367461 T^{3} + 1162035618775 T^{4} - 684320852667059 T^{5} + 351976137165513138 T^{6} - 684320852667059 p^{3} T^{7} + 1162035618775 p^{6} T^{8} - 1635367461 p^{9} T^{9} + 1938703 p^{12} T^{10} - 1704 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 67 | \( 1 - 2608162 T^{2} + 3205674513359 T^{4} - 2491312924906790680 T^{6} + \)\(13\!\cdots\!21\)\( T^{8} - \)\(58\!\cdots\!74\)\( T^{10} + \)\(19\!\cdots\!31\)\( T^{12} - \)\(58\!\cdots\!74\)\( p^{6} T^{14} + \)\(13\!\cdots\!21\)\( p^{12} T^{16} - 2491312924906790680 p^{18} T^{18} + 3205674513359 p^{24} T^{20} - 2608162 p^{30} T^{22} + p^{36} T^{24} \) | |
| 71 | \( 1 - 1935299 T^{2} + 2038830960730 T^{4} - 1525475006651412638 T^{6} + \)\(89\!\cdots\!49\)\( T^{8} - \)\(42\!\cdots\!91\)\( T^{10} + \)\(16\!\cdots\!92\)\( T^{12} - \)\(42\!\cdots\!91\)\( p^{6} T^{14} + \)\(89\!\cdots\!49\)\( p^{12} T^{16} - 1525475006651412638 p^{18} T^{18} + 2038830960730 p^{24} T^{20} - 1935299 p^{30} T^{22} + p^{36} T^{24} \) | |
| 73 | \( 1 - 2935638 T^{2} + 3898939340527 T^{4} - 3093494767983198216 T^{6} + \)\(16\!\cdots\!61\)\( T^{8} - \)\(68\!\cdots\!82\)\( T^{10} + \)\(25\!\cdots\!35\)\( T^{12} - \)\(68\!\cdots\!82\)\( p^{6} T^{14} + \)\(16\!\cdots\!61\)\( p^{12} T^{16} - 3093494767983198216 p^{18} T^{18} + 3898939340527 p^{24} T^{20} - 2935638 p^{30} T^{22} + p^{36} T^{24} \) | |
| 79 | \( ( 1 + 875 T + 2505744 T^{2} + 1612380868 T^{3} + 2680533156433 T^{4} + 1358001363523637 T^{5} + 1672684167171147060 T^{6} + 1358001363523637 p^{3} T^{7} + 2680533156433 p^{6} T^{8} + 1612380868 p^{9} T^{9} + 2505744 p^{12} T^{10} + 875 p^{15} T^{11} + p^{18} T^{12} )^{2} \) | |
| 83 | \( 1 - 3141070 T^{2} + 5512039858027 T^{4} - 6761964563893293052 T^{6} + \)\(64\!\cdots\!49\)\( T^{8} - \)\(48\!\cdots\!54\)\( T^{10} + \)\(30\!\cdots\!95\)\( T^{12} - \)\(48\!\cdots\!54\)\( p^{6} T^{14} + \)\(64\!\cdots\!49\)\( p^{12} T^{16} - 6761964563893293052 p^{18} T^{18} + 5512039858027 p^{24} T^{20} - 3141070 p^{30} T^{22} + p^{36} T^{24} \) | |
| 89 | \( 1 - 4477394 T^{2} + 9702807750839 T^{4} - 157611520547438048 p T^{6} + \)\(15\!\cdots\!13\)\( T^{8} - \)\(14\!\cdots\!50\)\( T^{10} + \)\(11\!\cdots\!03\)\( T^{12} - \)\(14\!\cdots\!50\)\( p^{6} T^{14} + \)\(15\!\cdots\!13\)\( p^{12} T^{16} - 157611520547438048 p^{19} T^{18} + 9702807750839 p^{24} T^{20} - 4477394 p^{30} T^{22} + p^{36} T^{24} \) | |
| 97 | \( 1 - 9021806 T^{2} + 37946584520559 T^{4} - 99120815383266274472 T^{6} + \)\(18\!\cdots\!09\)\( T^{8} - \)\(24\!\cdots\!66\)\( T^{10} + \)\(25\!\cdots\!63\)\( T^{12} - \)\(24\!\cdots\!66\)\( p^{6} T^{14} + \)\(18\!\cdots\!09\)\( p^{12} T^{16} - 99120815383266274472 p^{18} T^{18} + 37946584520559 p^{24} T^{20} - 9021806 p^{30} T^{22} + p^{36} T^{24} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.57256431375112625004671972238, −3.19532046482596558335048873384, −3.09582884559210710658992061698, −3.00893219717186423791829373622, −2.97280603610110326183609912313, −2.75274305867495526505873189976, −2.69370145396242740628232150211, −2.64962447015437579581122543019, −2.48058402945167771498317618514, −2.42960978256932254725045077543, −2.26474035264718660882074867350, −2.15679130988256361183891440009, −1.98952499002397111435051654659, −1.87532970908653512016328609221, −1.81992222531118537882535967640, −1.79135427229283974094067989148, −1.18605597315673265785970175965, −1.12619515800567375010360624990, −1.07929394871725255604005261470, −1.02267305121868424474347496869, −0.965570939166470091265330792405, −0.44156958468892323436473299486, −0.25488733965988766285706209639, −0.23764842748265696170937952456, −0.04768520933474828476461476316, 0.04768520933474828476461476316, 0.23764842748265696170937952456, 0.25488733965988766285706209639, 0.44156958468892323436473299486, 0.965570939166470091265330792405, 1.02267305121868424474347496869, 1.07929394871725255604005261470, 1.12619515800567375010360624990, 1.18605597315673265785970175965, 1.79135427229283974094067989148, 1.81992222531118537882535967640, 1.87532970908653512016328609221, 1.98952499002397111435051654659, 2.15679130988256361183891440009, 2.26474035264718660882074867350, 2.42960978256932254725045077543, 2.48058402945167771498317618514, 2.64962447015437579581122543019, 2.69370145396242740628232150211, 2.75274305867495526505873189976, 2.97280603610110326183609912313, 3.00893219717186423791829373622, 3.09582884559210710658992061698, 3.19532046482596558335048873384, 3.57256431375112625004671972238
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.