Dirichlet series
| L(s) = 1 | − 96·4-s − 10·5-s + 76·7-s + 1.46e3·9-s − 1.76e3·13-s + 5.37e3·16-s + 960·20-s − 4.76e3·23-s − 1.37e4·25-s − 7.29e3·28-s − 3.80e3·29-s − 760·35-s − 1.40e5·36-s − 1.46e4·45-s − 1.26e5·49-s + 1.69e5·52-s + 1.68e4·53-s + 620·59-s + 1.11e5·63-s − 2.29e5·64-s + 1.76e4·65-s + 3.00e4·67-s + 1.35e5·71-s − 5.37e4·80-s + 1.05e6·81-s + 68·83-s − 1.34e5·91-s + ⋯ |
| L(s) = 1 | − 3·4-s − 0.178·5-s + 0.586·7-s + 6.02·9-s − 2.89·13-s + 21/4·16-s + 0.536·20-s − 1.87·23-s − 4.39·25-s − 1.75·28-s − 0.840·29-s − 0.104·35-s − 18.0·36-s − 1.07·45-s − 7.54·49-s + 8.69·52-s + 0.823·53-s + 0.0231·59-s + 3.52·63-s − 7·64-s + 0.518·65-s + 0.817·67-s + 3.19·71-s − 0.939·80-s + 17.9·81-s + 0.00108·83-s − 1.69·91-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 29^{12}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 29^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.19819\times 10^{11}\) |
| Root analytic conductor: | \(3.04996\) |
| Motivic weight: | \(5\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 29^{12} ,\ ( \ : [5/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(3)\) | \(\approx\) | \(0.01954434830\) |
| \(L(\frac12)\) | \(\approx\) | \(0.01954434830\) |
| \(L(\frac{7}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p^{4} T^{2} )^{6} \) |
| 29 | \( 1 + 3808 T - 23834590 T^{2} - 66956320 p^{2} T^{3} + 21395960379 p^{3} T^{4} - 22084089920 p^{5} T^{5} - 999778041076 p^{7} T^{6} - 22084089920 p^{10} T^{7} + 21395960379 p^{13} T^{8} - 66956320 p^{17} T^{9} - 23834590 p^{20} T^{10} + 3808 p^{25} T^{11} + p^{30} T^{12} \) | |
| good | 3 | \( 1 - 1463 T^{2} + 360242 p T^{4} - 542498405 T^{6} + 209173073572 T^{8} - 7312489292155 p^{2} T^{10} + 214596890381992 p^{4} T^{12} - 7312489292155 p^{12} T^{14} + 209173073572 p^{20} T^{16} - 542498405 p^{30} T^{18} + 360242 p^{41} T^{20} - 1463 p^{50} T^{22} + p^{60} T^{24} \) |
| 5 | \( ( 1 + p T + 6902 T^{2} + 4843 p T^{3} + 21954968 T^{4} + 20078181 p T^{5} + 56579166004 T^{6} + 20078181 p^{6} T^{7} + 21954968 p^{10} T^{8} + 4843 p^{16} T^{9} + 6902 p^{20} T^{10} + p^{26} T^{11} + p^{30} T^{12} )^{2} \) | |
| 7 | \( ( 1 - 38 T + 65554 T^{2} - 253406 p T^{3} + 2088879951 T^{4} - 46550899452 T^{5} + 42743589948060 T^{6} - 46550899452 p^{5} T^{7} + 2088879951 p^{10} T^{8} - 253406 p^{16} T^{9} + 65554 p^{20} T^{10} - 38 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 11 | \( 1 - 44285 p T^{2} + 103675146054 T^{4} - 18059308866846845 T^{6} + \)\(35\!\cdots\!00\)\( T^{8} - \)\(55\!\cdots\!95\)\( T^{10} + \)\(76\!\cdots\!40\)\( T^{12} - \)\(55\!\cdots\!95\)\( p^{10} T^{14} + \)\(35\!\cdots\!00\)\( p^{20} T^{16} - 18059308866846845 p^{30} T^{18} + 103675146054 p^{40} T^{20} - 44285 p^{51} T^{22} + p^{60} T^{24} \) | |
| 13 | \( ( 1 + 883 T + 1072262 T^{2} + 495137309 T^{3} + 282896164116 T^{4} + 13046768616567 T^{5} + 11762434088788704 T^{6} + 13046768616567 p^{5} T^{7} + 282896164116 p^{10} T^{8} + 495137309 p^{15} T^{9} + 1072262 p^{20} T^{10} + 883 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 17 | \( 1 - 10081912 T^{2} + 47227775749146 T^{4} - \)\(14\!\cdots\!20\)\( T^{6} + \)\(31\!\cdots\!47\)\( T^{8} - \)\(58\!\cdots\!80\)\( T^{10} + \)\(90\!\cdots\!12\)\( T^{12} - \)\(58\!\cdots\!80\)\( p^{10} T^{14} + \)\(31\!\cdots\!47\)\( p^{20} T^{16} - \)\(14\!\cdots\!20\)\( p^{30} T^{18} + 47227775749146 p^{40} T^{20} - 10081912 p^{50} T^{22} + p^{60} T^{24} \) | |
| 19 | \( 1 - 11697240 T^{2} + 66016941765978 T^{4} - \)\(27\!\cdots\!36\)\( T^{6} + \)\(10\!\cdots\!91\)\( T^{8} - \)\(30\!\cdots\!24\)\( T^{10} + \)\(80\!\cdots\!60\)\( T^{12} - \)\(30\!\cdots\!24\)\( p^{10} T^{14} + \)\(10\!\cdots\!91\)\( p^{20} T^{16} - \)\(27\!\cdots\!36\)\( p^{30} T^{18} + 66016941765978 p^{40} T^{20} - 11697240 p^{50} T^{22} + p^{60} T^{24} \) | |
| 23 | \( ( 1 + 2382 T + 13637826 T^{2} + 11574349002 T^{3} + 111145418429679 T^{4} + 144798142045147116 T^{5} + \)\(10\!\cdots\!20\)\( T^{6} + 144798142045147116 p^{5} T^{7} + 111145418429679 p^{10} T^{8} + 11574349002 p^{15} T^{9} + 13637826 p^{20} T^{10} + 2382 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 31 | \( 1 - 281352015 T^{2} + 37857674462579982 T^{4} - \)\(32\!\cdots\!53\)\( T^{6} + \)\(19\!\cdots\!48\)\( T^{8} - \)\(82\!\cdots\!63\)\( T^{10} + \)\(27\!\cdots\!00\)\( T^{12} - \)\(82\!\cdots\!63\)\( p^{10} T^{14} + \)\(19\!\cdots\!48\)\( p^{20} T^{16} - \)\(32\!\cdots\!53\)\( p^{30} T^{18} + 37857674462579982 p^{40} T^{20} - 281352015 p^{50} T^{22} + p^{60} T^{24} \) | |
| 37 | \( 1 - 389094988 T^{2} + 79674512021554514 T^{4} - \)\(11\!\cdots\!20\)\( T^{6} + \)\(12\!\cdots\!35\)\( T^{8} - \)\(11\!\cdots\!00\)\( T^{10} + \)\(84\!\cdots\!00\)\( T^{12} - \)\(11\!\cdots\!00\)\( p^{10} T^{14} + \)\(12\!\cdots\!35\)\( p^{20} T^{16} - \)\(11\!\cdots\!20\)\( p^{30} T^{18} + 79674512021554514 p^{40} T^{20} - 389094988 p^{50} T^{22} + p^{60} T^{24} \) | |
| 41 | \( 1 - 591645912 T^{2} + 190105950435132090 T^{4} - \)\(43\!\cdots\!52\)\( T^{6} + \)\(79\!\cdots\!07\)\( T^{8} - \)\(11\!\cdots\!36\)\( T^{10} + \)\(14\!\cdots\!04\)\( T^{12} - \)\(11\!\cdots\!36\)\( p^{10} T^{14} + \)\(79\!\cdots\!07\)\( p^{20} T^{16} - \)\(43\!\cdots\!52\)\( p^{30} T^{18} + 190105950435132090 p^{40} T^{20} - 591645912 p^{50} T^{22} + p^{60} T^{24} \) | |
| 43 | \( 1 - 918372935 T^{2} + 386966810138932214 T^{4} - \)\(98\!\cdots\!25\)\( T^{6} + \)\(16\!\cdots\!60\)\( T^{8} - \)\(21\!\cdots\!75\)\( T^{10} + \)\(28\!\cdots\!00\)\( T^{12} - \)\(21\!\cdots\!75\)\( p^{10} T^{14} + \)\(16\!\cdots\!60\)\( p^{20} T^{16} - \)\(98\!\cdots\!25\)\( p^{30} T^{18} + 386966810138932214 p^{40} T^{20} - 918372935 p^{50} T^{22} + p^{60} T^{24} \) | |
| 47 | \( 1 - 1285566815 T^{2} + 937576996530230014 T^{4} - \)\(47\!\cdots\!25\)\( T^{6} + \)\(18\!\cdots\!60\)\( T^{8} - \)\(56\!\cdots\!75\)\( T^{10} + \)\(14\!\cdots\!00\)\( T^{12} - \)\(56\!\cdots\!75\)\( p^{10} T^{14} + \)\(18\!\cdots\!60\)\( p^{20} T^{16} - \)\(47\!\cdots\!25\)\( p^{30} T^{18} + 937576996530230014 p^{40} T^{20} - 1285566815 p^{50} T^{22} + p^{60} T^{24} \) | |
| 53 | \( ( 1 - 8421 T + 1513568510 T^{2} - 23783410193507 T^{3} + 1011011949473912356 T^{4} - \)\(22\!\cdots\!09\)\( T^{5} + \)\(46\!\cdots\!56\)\( T^{6} - \)\(22\!\cdots\!09\)\( p^{5} T^{7} + 1011011949473912356 p^{10} T^{8} - 23783410193507 p^{15} T^{9} + 1513568510 p^{20} T^{10} - 8421 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 59 | \( ( 1 - 310 T + 2887861250 T^{2} + 266285286470 T^{3} + 4135004951744690471 T^{4} + 72529048339705684740 T^{5} + \)\(36\!\cdots\!56\)\( T^{6} + 72529048339705684740 p^{5} T^{7} + 4135004951744690471 p^{10} T^{8} + 266285286470 p^{15} T^{9} + 2887861250 p^{20} T^{10} - 310 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 61 | \( 1 - 1760122776 T^{2} + 3189692639186996010 T^{4} - \)\(34\!\cdots\!88\)\( T^{6} + \)\(38\!\cdots\!23\)\( T^{8} - \)\(33\!\cdots\!96\)\( T^{10} + \)\(31\!\cdots\!52\)\( T^{12} - \)\(33\!\cdots\!96\)\( p^{10} T^{14} + \)\(38\!\cdots\!23\)\( p^{20} T^{16} - \)\(34\!\cdots\!88\)\( p^{30} T^{18} + 3189692639186996010 p^{40} T^{20} - 1760122776 p^{50} T^{22} + p^{60} T^{24} \) | |
| 67 | \( ( 1 - 15020 T + 5396878946 T^{2} - 36772180474516 T^{3} + 13804358833408434855 T^{4} - \)\(48\!\cdots\!52\)\( T^{5} + \)\(22\!\cdots\!88\)\( T^{6} - \)\(48\!\cdots\!52\)\( p^{5} T^{7} + 13804358833408434855 p^{10} T^{8} - 36772180474516 p^{15} T^{9} + 5396878946 p^{20} T^{10} - 15020 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 71 | \( ( 1 - 67920 T + 11600995562 T^{2} - 584571726514288 T^{3} + 53835252831361337503 T^{4} - \)\(20\!\cdots\!48\)\( T^{5} + \)\(13\!\cdots\!80\)\( T^{6} - \)\(20\!\cdots\!48\)\( p^{5} T^{7} + 53835252831361337503 p^{10} T^{8} - 584571726514288 p^{15} T^{9} + 11600995562 p^{20} T^{10} - 67920 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 73 | \( 1 - 13898145804 T^{2} + 95723285260862301858 T^{4} - \)\(43\!\cdots\!40\)\( T^{6} + \)\(15\!\cdots\!39\)\( T^{8} - \)\(42\!\cdots\!60\)\( T^{10} + \)\(96\!\cdots\!04\)\( T^{12} - \)\(42\!\cdots\!60\)\( p^{10} T^{14} + \)\(15\!\cdots\!39\)\( p^{20} T^{16} - \)\(43\!\cdots\!40\)\( p^{30} T^{18} + 95723285260862301858 p^{40} T^{20} - 13898145804 p^{50} T^{22} + p^{60} T^{24} \) | |
| 79 | \( 1 - 23335674183 T^{2} + \)\(26\!\cdots\!46\)\( T^{4} - \)\(19\!\cdots\!45\)\( T^{6} + \)\(10\!\cdots\!20\)\( T^{8} - \)\(44\!\cdots\!47\)\( T^{10} + \)\(15\!\cdots\!16\)\( T^{12} - \)\(44\!\cdots\!47\)\( p^{10} T^{14} + \)\(10\!\cdots\!20\)\( p^{20} T^{16} - \)\(19\!\cdots\!45\)\( p^{30} T^{18} + \)\(26\!\cdots\!46\)\( p^{40} T^{20} - 23335674183 p^{50} T^{22} + p^{60} T^{24} \) | |
| 83 | \( ( 1 - 34 T + 15124189634 T^{2} - 109420466663966 T^{3} + \)\(11\!\cdots\!83\)\( T^{4} - \)\(10\!\cdots\!84\)\( T^{5} + \)\(54\!\cdots\!52\)\( T^{6} - \)\(10\!\cdots\!84\)\( p^{5} T^{7} + \)\(11\!\cdots\!83\)\( p^{10} T^{8} - 109420466663966 p^{15} T^{9} + 15124189634 p^{20} T^{10} - 34 p^{25} T^{11} + p^{30} T^{12} )^{2} \) | |
| 89 | \( 1 - 30366806776 T^{2} + \)\(48\!\cdots\!30\)\( T^{4} - \)\(54\!\cdots\!32\)\( T^{6} + \)\(47\!\cdots\!67\)\( T^{8} - \)\(34\!\cdots\!96\)\( T^{10} + \)\(20\!\cdots\!12\)\( T^{12} - \)\(34\!\cdots\!96\)\( p^{10} T^{14} + \)\(47\!\cdots\!67\)\( p^{20} T^{16} - \)\(54\!\cdots\!32\)\( p^{30} T^{18} + \)\(48\!\cdots\!30\)\( p^{40} T^{20} - 30366806776 p^{50} T^{22} + p^{60} T^{24} \) | |
| 97 | \( 1 - 41978727288 T^{2} + \)\(86\!\cdots\!34\)\( T^{4} - \)\(12\!\cdots\!20\)\( T^{6} + \)\(14\!\cdots\!55\)\( T^{8} - \)\(14\!\cdots\!00\)\( T^{10} + \)\(12\!\cdots\!20\)\( T^{12} - \)\(14\!\cdots\!00\)\( p^{10} T^{14} + \)\(14\!\cdots\!55\)\( p^{20} T^{16} - \)\(12\!\cdots\!20\)\( p^{30} T^{18} + \)\(86\!\cdots\!34\)\( p^{40} T^{20} - 41978727288 p^{50} T^{22} + p^{60} T^{24} \) | |
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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
−4.60282842967247035469091862187, −4.49333557335208294285060000227, −4.20711834971872427754934053149, −3.98770125654400010815400217192, −3.96513562341249503794783371836, −3.93400347261498060540645278522, −3.77329605333118495688540788822, −3.69084269661171682305999785144, −3.51301288996285045414330016593, −3.28952528926704215481230064771, −3.16182766207791782239011137178, −2.80915789635731449714257479199, −2.42840935952768611896068815702, −2.22461184663562268281586417981, −2.01250725741293269634023256420, −1.94412048621356762943413855771, −1.89549441328227800052257697945, −1.57560691578711354176625199833, −1.41473158628959921653670091796, −1.31508592304719268361340880153, −1.20734971237976438190828028754, −0.64674230775203376295798870827, −0.51172263447107962510628386996, −0.26701230388504113274305724615, −0.02082808639041428451731844387, 0.02082808639041428451731844387, 0.26701230388504113274305724615, 0.51172263447107962510628386996, 0.64674230775203376295798870827, 1.20734971237976438190828028754, 1.31508592304719268361340880153, 1.41473158628959921653670091796, 1.57560691578711354176625199833, 1.89549441328227800052257697945, 1.94412048621356762943413855771, 2.01250725741293269634023256420, 2.22461184663562268281586417981, 2.42840935952768611896068815702, 2.80915789635731449714257479199, 3.16182766207791782239011137178, 3.28952528926704215481230064771, 3.51301288996285045414330016593, 3.69084269661171682305999785144, 3.77329605333118495688540788822, 3.93400347261498060540645278522, 3.96513562341249503794783371836, 3.98770125654400010815400217192, 4.20711834971872427754934053149, 4.49333557335208294285060000227, 4.60282842967247035469091862187
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.