Dirichlet series
| L(s) = 1 | − 6·2-s + 18·4-s − 6·5-s − 4·7-s − 32·8-s + 24·9-s + 36·10-s + 14·13-s + 24·14-s + 34·16-s + 2·17-s − 144·18-s + 14·19-s − 108·20-s + 56·23-s + 18·25-s − 84·26-s − 72·28-s + 60·29-s + 72·31-s − 4·32-s − 12·34-s + 24·35-s + 432·36-s − 66·37-s − 84·38-s + 192·40-s + ⋯ |
| L(s) = 1 | − 3·2-s + 9/2·4-s − 6/5·5-s − 4/7·7-s − 4·8-s + 8/3·9-s + 18/5·10-s + 1.07·13-s + 12/7·14-s + 17/8·16-s + 2/17·17-s − 8·18-s + 0.736·19-s − 5.39·20-s + 2.43·23-s + 0.719·25-s − 3.23·26-s − 2.57·28-s + 2.06·29-s + 2.32·31-s − 1/8·32-s − 0.352·34-s + 0.685·35-s + 12·36-s − 1.78·37-s − 2.21·38-s + 24/5·40-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(37^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(37^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.10265\) |
| Root analytic conductor: | \(1.00408\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 37^{12} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.2743294937\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2743294937\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 37 | \( 1 + 66 T + 4162 T^{2} + 199154 T^{3} + 215335 p T^{4} + 191172 p^{2} T^{5} + 214708 p^{3} T^{6} + 191172 p^{4} T^{7} + 215335 p^{5} T^{8} + 199154 p^{6} T^{9} + 4162 p^{8} T^{10} + 66 p^{10} T^{11} + p^{12} T^{12} \) |
| good | 2 | \( 1 + 3 p T + 9 p T^{2} + p^{5} T^{3} + 13 p T^{4} - 11 p^{2} T^{5} - 55 p^{2} T^{6} - 63 p^{3} T^{7} - 807 T^{8} - 441 p T^{9} - 103 p T^{10} + 345 p^{3} T^{11} + 263 p^{5} T^{12} + 345 p^{5} T^{13} - 103 p^{5} T^{14} - 441 p^{7} T^{15} - 807 p^{8} T^{16} - 63 p^{13} T^{17} - 55 p^{14} T^{18} - 11 p^{16} T^{19} + 13 p^{17} T^{20} + p^{23} T^{21} + 9 p^{21} T^{22} + 3 p^{23} T^{23} + p^{24} T^{24} \) |
| 3 | \( 1 - 8 p T^{2} + 442 T^{4} - 5590 T^{6} + 19697 p T^{8} - 580142 T^{10} + 566941 p^{2} T^{12} - 580142 p^{4} T^{14} + 19697 p^{9} T^{16} - 5590 p^{12} T^{18} + 442 p^{16} T^{20} - 8 p^{21} T^{22} + p^{24} T^{24} \) | |
| 5 | \( 1 + 6 T + 18 T^{2} + 54 p T^{3} + 1913 T^{4} + 1892 T^{5} + 13368 T^{6} + 186304 T^{7} - 431224 T^{8} - 6823228 T^{9} - 6658408 T^{10} - 125972648 T^{11} - 1433323744 T^{12} - 125972648 p^{2} T^{13} - 6658408 p^{4} T^{14} - 6823228 p^{6} T^{15} - 431224 p^{8} T^{16} + 186304 p^{10} T^{17} + 13368 p^{12} T^{18} + 1892 p^{14} T^{19} + 1913 p^{16} T^{20} + 54 p^{19} T^{21} + 18 p^{20} T^{22} + 6 p^{22} T^{23} + p^{24} T^{24} \) | |
| 7 | \( ( 1 + 2 T + 113 T^{2} + 18 T^{3} + 1011 p T^{4} + 7516 T^{5} + 417358 T^{6} + 7516 p^{2} T^{7} + 1011 p^{5} T^{8} + 18 p^{6} T^{9} + 113 p^{8} T^{10} + 2 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 11 | \( 1 - 568 T^{2} + 179482 T^{4} - 39959350 T^{6} + 7249986259 T^{8} - 1114607389614 T^{10} + 146730568724485 T^{12} - 1114607389614 p^{4} T^{14} + 7249986259 p^{8} T^{16} - 39959350 p^{12} T^{18} + 179482 p^{16} T^{20} - 568 p^{20} T^{22} + p^{24} T^{24} \) | |
| 13 | \( 1 - 14 T + 98 T^{2} - 2830 T^{3} + 1977 T^{4} + 630668 T^{5} - 5018648 T^{6} + 155012800 T^{7} - 1979069512 T^{8} - 9669589396 T^{9} + 99429769928 T^{10} - 3382366440384 T^{11} + 107370611618752 T^{12} - 3382366440384 p^{2} T^{13} + 99429769928 p^{4} T^{14} - 9669589396 p^{6} T^{15} - 1979069512 p^{8} T^{16} + 155012800 p^{10} T^{17} - 5018648 p^{12} T^{18} + 630668 p^{14} T^{19} + 1977 p^{16} T^{20} - 2830 p^{18} T^{21} + 98 p^{20} T^{22} - 14 p^{22} T^{23} + p^{24} T^{24} \) | |
| 17 | \( 1 - 2 T + 2 T^{2} - 3002 T^{3} - 196790 T^{4} + 874406 T^{5} + 3150770 T^{6} + 350111326 T^{7} + 19564834959 T^{8} - 48545502716 T^{9} + 9260089580 T^{10} - 10720819045644 T^{11} - 1687195162436724 T^{12} - 10720819045644 p^{2} T^{13} + 9260089580 p^{4} T^{14} - 48545502716 p^{6} T^{15} + 19564834959 p^{8} T^{16} + 350111326 p^{10} T^{17} + 3150770 p^{12} T^{18} + 874406 p^{14} T^{19} - 196790 p^{16} T^{20} - 3002 p^{18} T^{21} + 2 p^{20} T^{22} - 2 p^{22} T^{23} + p^{24} T^{24} \) | |
| 19 | \( 1 - 14 T + 98 T^{2} - 10102 T^{3} + 275722 T^{4} + 49146 T^{5} + 23316402 T^{6} - 1774162110 T^{7} - 17497848849 T^{8} + 835000045212 T^{9} + 1519736861868 T^{10} + 151879358088172 T^{11} - 9144006932525172 T^{12} + 151879358088172 p^{2} T^{13} + 1519736861868 p^{4} T^{14} + 835000045212 p^{6} T^{15} - 17497848849 p^{8} T^{16} - 1774162110 p^{10} T^{17} + 23316402 p^{12} T^{18} + 49146 p^{14} T^{19} + 275722 p^{16} T^{20} - 10102 p^{18} T^{21} + 98 p^{20} T^{22} - 14 p^{22} T^{23} + p^{24} T^{24} \) | |
| 23 | \( 1 - 56 T + 1568 T^{2} - 58240 T^{3} + 1750805 T^{4} - 28505216 T^{5} + 546978656 T^{6} - 12897170080 T^{7} + 49409450876 T^{8} + 2640937231392 T^{9} - 25614614687072 T^{10} + 1895696020371192 T^{11} - 85259645088616228 T^{12} + 1895696020371192 p^{2} T^{13} - 25614614687072 p^{4} T^{14} + 2640937231392 p^{6} T^{15} + 49409450876 p^{8} T^{16} - 12897170080 p^{10} T^{17} + 546978656 p^{12} T^{18} - 28505216 p^{14} T^{19} + 1750805 p^{16} T^{20} - 58240 p^{18} T^{21} + 1568 p^{20} T^{22} - 56 p^{22} T^{23} + p^{24} T^{24} \) | |
| 29 | \( 1 - 60 T + 1800 T^{2} - 76260 T^{3} + 3522317 T^{4} - 3175412 p T^{5} + 2092840080 T^{6} - 68498506208 T^{7} + 1051137951764 T^{8} + 13534145006492 T^{9} - 612044126230648 T^{10} + 30769026299619880 T^{11} - 1339171925891845540 T^{12} + 30769026299619880 p^{2} T^{13} - 612044126230648 p^{4} T^{14} + 13534145006492 p^{6} T^{15} + 1051137951764 p^{8} T^{16} - 68498506208 p^{10} T^{17} + 2092840080 p^{12} T^{18} - 3175412 p^{15} T^{19} + 3522317 p^{16} T^{20} - 76260 p^{18} T^{21} + 1800 p^{20} T^{22} - 60 p^{22} T^{23} + p^{24} T^{24} \) | |
| 31 | \( 1 - 72 T + 2592 T^{2} - 66112 T^{3} + 3862457 T^{4} - 223788600 T^{5} + 8286688928 T^{6} - 223207908928 T^{7} + 7964791705256 T^{8} - 359858308245960 T^{9} + 13055468456937408 T^{10} - 348753368966413784 T^{11} + 9315533212577603808 T^{12} - 348753368966413784 p^{2} T^{13} + 13055468456937408 p^{4} T^{14} - 359858308245960 p^{6} T^{15} + 7964791705256 p^{8} T^{16} - 223207908928 p^{10} T^{17} + 8286688928 p^{12} T^{18} - 223788600 p^{14} T^{19} + 3862457 p^{16} T^{20} - 66112 p^{18} T^{21} + 2592 p^{20} T^{22} - 72 p^{22} T^{23} + p^{24} T^{24} \) | |
| 41 | \( 1 - 13516 T^{2} + 2070338 p T^{4} - 336419462698 T^{6} + 967603772672547 T^{8} - 2186823060207879770 T^{10} + \)\(40\!\cdots\!01\)\( T^{12} - 2186823060207879770 p^{4} T^{14} + 967603772672547 p^{8} T^{16} - 336419462698 p^{12} T^{18} + 2070338 p^{17} T^{20} - 13516 p^{20} T^{22} + p^{24} T^{24} \) | |
| 43 | \( 1 - 70 T + 2450 T^{2} - 102126 T^{3} + 6053642 T^{4} - 303352222 T^{5} + 11618092578 T^{6} - 477914249878 T^{7} + 28313200939951 T^{8} - 1517599348789044 T^{9} + 61351594901493324 T^{10} - 2760177618670072100 T^{11} + \)\(12\!\cdots\!68\)\( T^{12} - 2760177618670072100 p^{2} T^{13} + 61351594901493324 p^{4} T^{14} - 1517599348789044 p^{6} T^{15} + 28313200939951 p^{8} T^{16} - 477914249878 p^{10} T^{17} + 11618092578 p^{12} T^{18} - 303352222 p^{14} T^{19} + 6053642 p^{16} T^{20} - 102126 p^{18} T^{21} + 2450 p^{20} T^{22} - 70 p^{22} T^{23} + p^{24} T^{24} \) | |
| 47 | \( ( 1 + 192 T + 27473 T^{2} + 2630660 T^{3} + 207915815 T^{4} + 12821350100 T^{5} + 672201407282 T^{6} + 12821350100 p^{2} T^{7} + 207915815 p^{4} T^{8} + 2630660 p^{6} T^{9} + 27473 p^{8} T^{10} + 192 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 53 | \( ( 1 + 28 T + 13275 T^{2} + 4416 p T^{3} + 77949657 T^{4} + 934896716 T^{5} + 273502311914 T^{6} + 934896716 p^{2} T^{7} + 77949657 p^{4} T^{8} + 4416 p^{7} T^{9} + 13275 p^{8} T^{10} + 28 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 59 | \( 1 - 184 T + 16928 T^{2} - 1662392 T^{3} + 155319838 T^{4} - 9347403288 T^{5} + 472441568160 T^{6} - 25069376532632 T^{7} + 3503812355437 p T^{8} + 89482902104177808 T^{9} - 7981736308771508160 T^{10} + \)\(68\!\cdots\!52\)\( T^{11} - \)\(51\!\cdots\!08\)\( T^{12} + \)\(68\!\cdots\!52\)\( p^{2} T^{13} - 7981736308771508160 p^{4} T^{14} + 89482902104177808 p^{6} T^{15} + 3503812355437 p^{9} T^{16} - 25069376532632 p^{10} T^{17} + 472441568160 p^{12} T^{18} - 9347403288 p^{14} T^{19} + 155319838 p^{16} T^{20} - 1662392 p^{18} T^{21} + 16928 p^{20} T^{22} - 184 p^{22} T^{23} + p^{24} T^{24} \) | |
| 61 | \( 1 - 132 T + 8712 T^{2} - 3028 p T^{3} + 15603369 T^{4} - 2128893420 T^{5} + 162135903344 T^{6} - 2795652825552 T^{7} - 158155642105360 T^{8} + 23161025581391524 T^{9} - 1426796711784843768 T^{10} + \)\(10\!\cdots\!52\)\( T^{11} - \)\(52\!\cdots\!64\)\( T^{12} + \)\(10\!\cdots\!52\)\( p^{2} T^{13} - 1426796711784843768 p^{4} T^{14} + 23161025581391524 p^{6} T^{15} - 158155642105360 p^{8} T^{16} - 2795652825552 p^{10} T^{17} + 162135903344 p^{12} T^{18} - 2128893420 p^{14} T^{19} + 15603369 p^{16} T^{20} - 3028 p^{19} T^{21} + 8712 p^{20} T^{22} - 132 p^{22} T^{23} + p^{24} T^{24} \) | |
| 67 | \( 1 - 26278 T^{2} + 354910729 T^{4} - 3284724101144 T^{6} + 23435236000970280 T^{8} - \)\(13\!\cdots\!54\)\( T^{10} + \)\(66\!\cdots\!12\)\( T^{12} - \)\(13\!\cdots\!54\)\( p^{4} T^{14} + 23435236000970280 p^{8} T^{16} - 3284724101144 p^{12} T^{18} + 354910729 p^{16} T^{20} - 26278 p^{20} T^{22} + p^{24} T^{24} \) | |
| 71 | \( ( 1 - 34 T + 9439 T^{2} - 630878 T^{3} + 77276687 T^{4} - 5460524096 T^{5} + 389697885202 T^{6} - 5460524096 p^{2} T^{7} + 77276687 p^{4} T^{8} - 630878 p^{6} T^{9} + 9439 p^{8} T^{10} - 34 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 73 | \( 1 - 25156 T^{2} + 399766714 T^{4} - 61840461858 p T^{6} + 39860433974871515 T^{8} - \)\(28\!\cdots\!14\)\( T^{10} + \)\(16\!\cdots\!53\)\( T^{12} - \)\(28\!\cdots\!14\)\( p^{4} T^{14} + 39860433974871515 p^{8} T^{16} - 61840461858 p^{13} T^{18} + 399766714 p^{16} T^{20} - 25156 p^{20} T^{22} + p^{24} T^{24} \) | |
| 79 | \( 1 + 2 T + 2 T^{2} - 570598 T^{3} - 25004591 T^{4} + 4218359272 T^{5} + 171277766528 T^{6} + 22643439587856 T^{7} - 1935501742475592 T^{8} - 145865388936686104 T^{9} - 28153143079563872 T^{10} - 48856127446563677784 T^{11} + \)\(10\!\cdots\!40\)\( T^{12} - 48856127446563677784 p^{2} T^{13} - 28153143079563872 p^{4} T^{14} - 145865388936686104 p^{6} T^{15} - 1935501742475592 p^{8} T^{16} + 22643439587856 p^{10} T^{17} + 171277766528 p^{12} T^{18} + 4218359272 p^{14} T^{19} - 25004591 p^{16} T^{20} - 570598 p^{18} T^{21} + 2 p^{20} T^{22} + 2 p^{22} T^{23} + p^{24} T^{24} \) | |
| 83 | \( ( 1 - 54 T + 14591 T^{2} - 1148858 T^{3} + 179308903 T^{4} - 11564668576 T^{5} + 1380114658530 T^{6} - 11564668576 p^{2} T^{7} + 179308903 p^{4} T^{8} - 1148858 p^{6} T^{9} + 14591 p^{8} T^{10} - 54 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 89 | \( 1 - 278 T + 38642 T^{2} - 4994710 T^{3} + 656252254 T^{4} - 63973702942 T^{5} + 4899353810858 T^{6} - 377640048229086 T^{7} + 25307539321784895 T^{8} - 825804955131636844 T^{9} - 31052279118941181148 T^{10} + \)\(64\!\cdots\!96\)\( T^{11} - \)\(61\!\cdots\!36\)\( T^{12} + \)\(64\!\cdots\!96\)\( p^{2} T^{13} - 31052279118941181148 p^{4} T^{14} - 825804955131636844 p^{6} T^{15} + 25307539321784895 p^{8} T^{16} - 377640048229086 p^{10} T^{17} + 4899353810858 p^{12} T^{18} - 63973702942 p^{14} T^{19} + 656252254 p^{16} T^{20} - 4994710 p^{18} T^{21} + 38642 p^{20} T^{22} - 278 p^{22} T^{23} + p^{24} T^{24} \) | |
| 97 | \( 1 + 244 T + 29768 T^{2} + 3616340 T^{3} + 495367114 T^{4} + 57191622468 T^{5} + 5747625130440 T^{6} + 641978884941316 T^{7} + 79138223942129935 T^{8} + 8370516493159120008 T^{9} + \)\(79\!\cdots\!64\)\( T^{10} + \)\(83\!\cdots\!56\)\( T^{11} + \)\(86\!\cdots\!56\)\( T^{12} + \)\(83\!\cdots\!56\)\( p^{2} T^{13} + \)\(79\!\cdots\!64\)\( p^{4} T^{14} + 8370516493159120008 p^{6} T^{15} + 79138223942129935 p^{8} T^{16} + 641978884941316 p^{10} T^{17} + 5747625130440 p^{12} T^{18} + 57191622468 p^{14} T^{19} + 495367114 p^{16} T^{20} + 3616340 p^{18} T^{21} + 29768 p^{20} T^{22} + 244 p^{22} T^{23} + p^{24} T^{24} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.33634479759904947366924745360, −6.28772541155145700474865303869, −6.11254517303102304633034296900, −5.90382030488663769854483882434, −5.78938299904279924002106091267, −5.35330861191263305372052125036, −5.00877821538763693485688733946, −4.96829853886162318994776492005, −4.86883614315670739913294515655, −4.85106509947345654662408430694, −4.82004970103758044785080306114, −4.63228361283962498269245408002, −4.29883622849917802003891067842, −3.91705442343832156249502095451, −3.86277696300865977899606637956, −3.58542025964866390961872243458, −3.36705816911608493574357440622, −3.20224590492805940480038951598, −3.18289942557877272407931282772, −2.88123668655514617087825592058, −2.41324746889312974751465145466, −1.77491268114193290212744434481, −1.71556156900858901497764893443, −1.10160782993406000975875803797, −1.02756784431474061476768627988, 1.02756784431474061476768627988, 1.10160782993406000975875803797, 1.71556156900858901497764893443, 1.77491268114193290212744434481, 2.41324746889312974751465145466, 2.88123668655514617087825592058, 3.18289942557877272407931282772, 3.20224590492805940480038951598, 3.36705816911608493574357440622, 3.58542025964866390961872243458, 3.86277696300865977899606637956, 3.91705442343832156249502095451, 4.29883622849917802003891067842, 4.63228361283962498269245408002, 4.82004970103758044785080306114, 4.85106509947345654662408430694, 4.86883614315670739913294515655, 4.96829853886162318994776492005, 5.00877821538763693485688733946, 5.35330861191263305372052125036, 5.78938299904279924002106091267, 5.90382030488663769854483882434, 6.11254517303102304633034296900, 6.28772541155145700474865303869, 6.33634479759904947366924745360
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.