Dirichlet series
| L(s) = 1 | + 15·2-s + 124·3-s + 81·4-s − 3.75e3·5-s + 1.86e3·6-s + 4.87e3·8-s + 4.18e3·9-s − 5.62e4·10-s − 4.77e4·11-s + 1.00e4·12-s − 1.02e5·13-s − 4.65e5·15-s + 2.78e5·16-s + 3.84e4·17-s + 6.27e4·18-s − 3.61e5·19-s − 3.03e5·20-s − 7.16e5·22-s + 6.97e5·23-s + 6.04e5·24-s + 8.20e6·25-s − 1.53e6·26-s − 1.48e6·27-s + 1.60e7·29-s − 6.97e6·30-s − 1.36e6·31-s + 1.01e6·32-s + ⋯ |
| L(s) = 1 | + 0.662·2-s + 0.883·3-s + 0.158·4-s − 2.68·5-s + 0.585·6-s + 0.420·8-s + 0.212·9-s − 1.77·10-s − 0.984·11-s + 0.139·12-s − 0.992·13-s − 2.37·15-s + 1.06·16-s + 0.111·17-s + 0.140·18-s − 0.635·19-s − 0.424·20-s − 0.652·22-s + 0.519·23-s + 0.372·24-s + 21/5·25-s − 0.657·26-s − 0.536·27-s + 4.20·29-s − 1.57·30-s − 0.265·31-s + 0.170·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(10-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{6} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+9/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(12\) |
| Conductor: | \(5^{6} \cdot 7^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.03665\times 10^{12}\) |
| Root analytic conductor: | \(11.2331\) |
| Motivic weight: | \(9\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((12,\ 5^{6} \cdot 7^{12} ,\ ( \ : [9/2]^{6} ),\ 1 )\) |
Particular Values
| \(L(5)\) | \(\approx\) | \(20.47791313\) |
| \(L(\frac12)\) | \(\approx\) | \(20.47791313\) |
| \(L(\frac{11}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 + p^{4} T )^{6} \) |
| 7 | \( 1 \) | |
| good | 2 | \( 1 - 15 T + 9 p^{4} T^{2} - 2911 p T^{3} - 32415 p^{2} T^{4} + 152505 p^{5} T^{5} + 701781 p^{6} T^{6} + 152505 p^{14} T^{7} - 32415 p^{20} T^{8} - 2911 p^{28} T^{9} + 9 p^{40} T^{10} - 15 p^{45} T^{11} + p^{54} T^{12} \) |
| 3 | \( 1 - 124 T + 11192 T^{2} + 280 p^{7} T^{3} - 5218624 p^{2} T^{4} - 2097956 p^{4} T^{5} + 65003194166 p^{4} T^{6} - 2097956 p^{13} T^{7} - 5218624 p^{20} T^{8} + 280 p^{34} T^{9} + 11192 p^{36} T^{10} - 124 p^{45} T^{11} + p^{54} T^{12} \) | |
| 11 | \( 1 + 47796 T + 8379830664 T^{2} + 270782212098536 T^{3} + 35489011284799449744 T^{4} + \)\(95\!\cdots\!20\)\( T^{5} + \)\(10\!\cdots\!58\)\( T^{6} + \)\(95\!\cdots\!20\)\( p^{9} T^{7} + 35489011284799449744 p^{18} T^{8} + 270782212098536 p^{27} T^{9} + 8379830664 p^{36} T^{10} + 47796 p^{45} T^{11} + p^{54} T^{12} \) | |
| 13 | \( 1 + 102168 T + 43775548728 T^{2} + 3697743349958732 T^{3} + \)\(90\!\cdots\!36\)\( T^{4} + \)\(63\!\cdots\!40\)\( T^{5} + \)\(11\!\cdots\!26\)\( T^{6} + \)\(63\!\cdots\!40\)\( p^{9} T^{7} + \)\(90\!\cdots\!36\)\( p^{18} T^{8} + 3697743349958732 p^{27} T^{9} + 43775548728 p^{36} T^{10} + 102168 p^{45} T^{11} + p^{54} T^{12} \) | |
| 17 | \( 1 - 38472 T + 422077582368 T^{2} - 57913193733474572 T^{3} + \)\(81\!\cdots\!72\)\( T^{4} - \)\(17\!\cdots\!08\)\( T^{5} + \)\(10\!\cdots\!62\)\( T^{6} - \)\(17\!\cdots\!08\)\( p^{9} T^{7} + \)\(81\!\cdots\!72\)\( p^{18} T^{8} - 57913193733474572 p^{27} T^{9} + 422077582368 p^{36} T^{10} - 38472 p^{45} T^{11} + p^{54} T^{12} \) | |
| 19 | \( 1 + 361056 T + 966857741034 T^{2} + 352940649239794944 T^{3} + \)\(59\!\cdots\!75\)\( T^{4} + \)\(96\!\cdots\!12\)\( p T^{5} + \)\(22\!\cdots\!40\)\( T^{6} + \)\(96\!\cdots\!12\)\( p^{10} T^{7} + \)\(59\!\cdots\!75\)\( p^{18} T^{8} + 352940649239794944 p^{27} T^{9} + 966857741034 p^{36} T^{10} + 361056 p^{45} T^{11} + p^{54} T^{12} \) | |
| 23 | \( 1 - 697032 T + 3741541369650 T^{2} - 1347501248298667768 T^{3} + \)\(50\!\cdots\!51\)\( T^{4} + \)\(45\!\cdots\!24\)\( T^{5} + \)\(50\!\cdots\!84\)\( T^{6} + \)\(45\!\cdots\!24\)\( p^{9} T^{7} + \)\(50\!\cdots\!51\)\( p^{18} T^{8} - 1347501248298667768 p^{27} T^{9} + 3741541369650 p^{36} T^{10} - 697032 p^{45} T^{11} + p^{54} T^{12} \) | |
| 29 | \( 1 - 552696 p T + 166361093577864 T^{2} - \)\(12\!\cdots\!76\)\( T^{3} + \)\(70\!\cdots\!40\)\( T^{4} - \)\(33\!\cdots\!12\)\( T^{5} + \)\(13\!\cdots\!30\)\( T^{6} - \)\(33\!\cdots\!12\)\( p^{9} T^{7} + \)\(70\!\cdots\!40\)\( p^{18} T^{8} - \)\(12\!\cdots\!76\)\( p^{27} T^{9} + 166361093577864 p^{36} T^{10} - 552696 p^{46} T^{11} + p^{54} T^{12} \) | |
| 31 | \( 1 + 1362912 T + 81684677462154 T^{2} + \)\(24\!\cdots\!40\)\( T^{3} + \)\(35\!\cdots\!95\)\( T^{4} + \)\(13\!\cdots\!60\)\( T^{5} + \)\(10\!\cdots\!84\)\( T^{6} + \)\(13\!\cdots\!60\)\( p^{9} T^{7} + \)\(35\!\cdots\!95\)\( p^{18} T^{8} + \)\(24\!\cdots\!40\)\( p^{27} T^{9} + 81684677462154 p^{36} T^{10} + 1362912 p^{45} T^{11} + p^{54} T^{12} \) | |
| 37 | \( 1 + 3912924 T + 195045151243146 T^{2} + \)\(17\!\cdots\!52\)\( T^{3} + \)\(41\!\cdots\!39\)\( T^{4} + \)\(39\!\cdots\!60\)\( T^{5} + \)\(42\!\cdots\!48\)\( T^{6} + \)\(39\!\cdots\!60\)\( p^{9} T^{7} + \)\(41\!\cdots\!39\)\( p^{18} T^{8} + \)\(17\!\cdots\!52\)\( p^{27} T^{9} + 195045151243146 p^{36} T^{10} + 3912924 p^{45} T^{11} + p^{54} T^{12} \) | |
| 41 | \( 1 + 22452756 T + 1622036797695018 T^{2} + \)\(28\!\cdots\!00\)\( T^{3} + \)\(11\!\cdots\!95\)\( T^{4} + \)\(16\!\cdots\!44\)\( T^{5} + \)\(45\!\cdots\!32\)\( T^{6} + \)\(16\!\cdots\!44\)\( p^{9} T^{7} + \)\(11\!\cdots\!95\)\( p^{18} T^{8} + \)\(28\!\cdots\!00\)\( p^{27} T^{9} + 1622036797695018 p^{36} T^{10} + 22452756 p^{45} T^{11} + p^{54} T^{12} \) | |
| 43 | \( 1 + 29998992 T + 2380551033522234 T^{2} + \)\(60\!\cdots\!60\)\( T^{3} + \)\(26\!\cdots\!11\)\( T^{4} + \)\(54\!\cdots\!52\)\( T^{5} + \)\(17\!\cdots\!48\)\( T^{6} + \)\(54\!\cdots\!52\)\( p^{9} T^{7} + \)\(26\!\cdots\!11\)\( p^{18} T^{8} + \)\(60\!\cdots\!60\)\( p^{27} T^{9} + 2380551033522234 p^{36} T^{10} + 29998992 p^{45} T^{11} + p^{54} T^{12} \) | |
| 47 | \( 1 - 121271508 T + 9406050408578688 T^{2} - \)\(46\!\cdots\!04\)\( T^{3} + \)\(18\!\cdots\!00\)\( T^{4} - \)\(56\!\cdots\!04\)\( T^{5} + \)\(18\!\cdots\!86\)\( T^{6} - \)\(56\!\cdots\!04\)\( p^{9} T^{7} + \)\(18\!\cdots\!00\)\( p^{18} T^{8} - \)\(46\!\cdots\!04\)\( p^{27} T^{9} + 9406050408578688 p^{36} T^{10} - 121271508 p^{45} T^{11} + p^{54} T^{12} \) | |
| 53 | \( 1 + 20308596 T + 5609660587202082 T^{2} + \)\(89\!\cdots\!80\)\( T^{3} + \)\(39\!\cdots\!59\)\( T^{4} + \)\(49\!\cdots\!56\)\( T^{5} + \)\(12\!\cdots\!84\)\( T^{6} + \)\(49\!\cdots\!56\)\( p^{9} T^{7} + \)\(39\!\cdots\!59\)\( p^{18} T^{8} + \)\(89\!\cdots\!80\)\( p^{27} T^{9} + 5609660587202082 p^{36} T^{10} + 20308596 p^{45} T^{11} + p^{54} T^{12} \) | |
| 59 | \( 1 + 120280392 T + 26534507921192850 T^{2} + \)\(30\!\cdots\!60\)\( T^{3} + \)\(44\!\cdots\!31\)\( T^{4} + \)\(40\!\cdots\!60\)\( T^{5} + \)\(45\!\cdots\!56\)\( T^{6} + \)\(40\!\cdots\!60\)\( p^{9} T^{7} + \)\(44\!\cdots\!31\)\( p^{18} T^{8} + \)\(30\!\cdots\!60\)\( p^{27} T^{9} + 26534507921192850 p^{36} T^{10} + 120280392 p^{45} T^{11} + p^{54} T^{12} \) | |
| 61 | \( 1 - 87693540 T + 25065803466079746 T^{2} - \)\(18\!\cdots\!12\)\( T^{3} + \)\(47\!\cdots\!23\)\( T^{4} - \)\(34\!\cdots\!48\)\( T^{5} + \)\(69\!\cdots\!60\)\( T^{6} - \)\(34\!\cdots\!48\)\( p^{9} T^{7} + \)\(47\!\cdots\!23\)\( p^{18} T^{8} - \)\(18\!\cdots\!12\)\( p^{27} T^{9} + 25065803466079746 p^{36} T^{10} - 87693540 p^{45} T^{11} + p^{54} T^{12} \) | |
| 67 | \( 1 - 495050664 T + 239003350816184034 T^{2} - \)\(71\!\cdots\!80\)\( T^{3} + \)\(19\!\cdots\!83\)\( T^{4} - \)\(40\!\cdots\!64\)\( T^{5} + \)\(74\!\cdots\!48\)\( T^{6} - \)\(40\!\cdots\!64\)\( p^{9} T^{7} + \)\(19\!\cdots\!83\)\( p^{18} T^{8} - \)\(71\!\cdots\!80\)\( p^{27} T^{9} + 239003350816184034 p^{36} T^{10} - 495050664 p^{45} T^{11} + p^{54} T^{12} \) | |
| 71 | \( 1 - 253762512 T + 152405832326709738 T^{2} - \)\(48\!\cdots\!44\)\( p T^{3} + \)\(86\!\cdots\!79\)\( T^{4} - \)\(20\!\cdots\!24\)\( T^{5} + \)\(34\!\cdots\!84\)\( T^{6} - \)\(20\!\cdots\!24\)\( p^{9} T^{7} + \)\(86\!\cdots\!79\)\( p^{18} T^{8} - \)\(48\!\cdots\!44\)\( p^{28} T^{9} + 152405832326709738 p^{36} T^{10} - 253762512 p^{45} T^{11} + p^{54} T^{12} \) | |
| 73 | \( 1 + 187195308 T + 219035051028369810 T^{2} + \)\(34\!\cdots\!92\)\( T^{3} + \)\(24\!\cdots\!63\)\( T^{4} + \)\(33\!\cdots\!04\)\( T^{5} + \)\(17\!\cdots\!36\)\( T^{6} + \)\(33\!\cdots\!04\)\( p^{9} T^{7} + \)\(24\!\cdots\!63\)\( p^{18} T^{8} + \)\(34\!\cdots\!92\)\( p^{27} T^{9} + 219035051028369810 p^{36} T^{10} + 187195308 p^{45} T^{11} + p^{54} T^{12} \) | |
| 79 | \( 1 - 831079500 T + 689785417883020464 T^{2} - \)\(34\!\cdots\!44\)\( T^{3} + \)\(18\!\cdots\!80\)\( T^{4} - \)\(71\!\cdots\!08\)\( T^{5} + \)\(28\!\cdots\!50\)\( T^{6} - \)\(71\!\cdots\!08\)\( p^{9} T^{7} + \)\(18\!\cdots\!80\)\( p^{18} T^{8} - \)\(34\!\cdots\!44\)\( p^{27} T^{9} + 689785417883020464 p^{36} T^{10} - 831079500 p^{45} T^{11} + p^{54} T^{12} \) | |
| 83 | \( 1 - 767650536 T + 1190415298294226274 T^{2} - \)\(67\!\cdots\!68\)\( T^{3} + \)\(68\!\cdots\!89\)\( p T^{4} - \)\(24\!\cdots\!28\)\( T^{5} + \)\(14\!\cdots\!32\)\( T^{6} - \)\(24\!\cdots\!28\)\( p^{9} T^{7} + \)\(68\!\cdots\!89\)\( p^{19} T^{8} - \)\(67\!\cdots\!68\)\( p^{27} T^{9} + 1190415298294226274 p^{36} T^{10} - 767650536 p^{45} T^{11} + p^{54} T^{12} \) | |
| 89 | \( 1 + 582579684 T + 878699529449962794 T^{2} + \)\(31\!\cdots\!36\)\( T^{3} + \)\(36\!\cdots\!75\)\( T^{4} + \)\(12\!\cdots\!32\)\( T^{5} + \)\(14\!\cdots\!80\)\( T^{6} + \)\(12\!\cdots\!32\)\( p^{9} T^{7} + \)\(36\!\cdots\!75\)\( p^{18} T^{8} + \)\(31\!\cdots\!36\)\( p^{27} T^{9} + 878699529449962794 p^{36} T^{10} + 582579684 p^{45} T^{11} + p^{54} T^{12} \) | |
| 97 | \( 1 - 1184506872 T + 3794394595772867376 T^{2} - \)\(34\!\cdots\!24\)\( T^{3} + \)\(62\!\cdots\!12\)\( T^{4} - \)\(44\!\cdots\!16\)\( T^{5} + \)\(59\!\cdots\!78\)\( T^{6} - \)\(44\!\cdots\!16\)\( p^{9} T^{7} + \)\(62\!\cdots\!12\)\( p^{18} T^{8} - \)\(34\!\cdots\!24\)\( p^{27} T^{9} + 3794394595772867376 p^{36} T^{10} - 1184506872 p^{45} T^{11} + p^{54} T^{12} \) | |
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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
−4.88892475965948432729614973156, −4.67253689967575377668159876059, −4.57766618182709075197204678453, −4.39318789323935126147117898917, −4.24386033592782328504171826806, −4.15549908477214693991186791039, −3.66306381491149246963765820548, −3.60980859252030790795539139060, −3.52831241183843154712829381368, −3.27845567835966699837448118075, −3.14486665482148570902099950924, −2.99930232530765723239217395708, −2.66293787588151903941121462924, −2.54401093013199224086595471696, −2.30982448605790971146338017858, −2.25246324234638711989740714389, −1.89589620920425496551959905172, −1.54957443748032929990365395369, −1.53847835124681322768619383783, −1.07520255408569780307108961965, −0.71964061932215793592831510983, −0.65402242424542385590256043894, −0.60814569042436757102361192179, −0.46361497697710575950871790309, −0.28407804969798824765291736644, 0.28407804969798824765291736644, 0.46361497697710575950871790309, 0.60814569042436757102361192179, 0.65402242424542385590256043894, 0.71964061932215793592831510983, 1.07520255408569780307108961965, 1.53847835124681322768619383783, 1.54957443748032929990365395369, 1.89589620920425496551959905172, 2.25246324234638711989740714389, 2.30982448605790971146338017858, 2.54401093013199224086595471696, 2.66293787588151903941121462924, 2.99930232530765723239217395708, 3.14486665482148570902099950924, 3.27845567835966699837448118075, 3.52831241183843154712829381368, 3.60980859252030790795539139060, 3.66306381491149246963765820548, 4.15549908477214693991186791039, 4.24386033592782328504171826806, 4.39318789323935126147117898917, 4.57766618182709075197204678453, 4.67253689967575377668159876059, 4.88892475965948432729614973156
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.