Dirichlet series
| L(s) = 1 | − 8.46e3·2-s − 8.46e7·4-s + 9.91e11·8-s − 2.59e12·9-s + 3.95e13·11-s + 1.64e15·16-s + 2.19e16·18-s − 3.34e17·22-s + 7.71e16·23-s − 6.39e17·25-s + 8.69e18·29-s − 4.92e19·32-s + 2.19e20·36-s + 1.51e20·37-s − 3.80e20·43-s − 3.35e21·44-s − 6.53e20·46-s + 5.41e21·50-s + 1.77e22·53-s − 7.35e22·58-s + 6.01e22·64-s + 1.48e23·67-s + 5.62e23·71-s − 2.57e24·72-s − 1.28e24·74-s + 1.48e24·79-s + 1.65e24·81-s + ⋯ |
| L(s) = 1 | − 1.46·2-s − 2.52·4-s + 5.10·8-s − 3.06·9-s + 3.80·11-s + 1.45·16-s + 4.47·18-s − 5.55·22-s + 0.734·23-s − 2.14·25-s + 4.56·29-s − 7.54·32-s + 7.72·36-s + 3.79·37-s − 1.45·43-s − 9.59·44-s − 1.07·46-s + 3.13·50-s + 4.96·53-s − 6.66·58-s + 1.59·64-s + 2.21·67-s + 4.06·71-s − 15.6·72-s − 5.53·74-s + 2.83·79-s + 2.30·81-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(26-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(7^{24}\right)^{s/2} \, \Gamma_{\C}(s+25/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(7^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.84874\times 10^{27}\) |
| Root analytic conductor: | \(13.9297\) |
| Motivic weight: | \(25\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 7^{24} ,\ ( \ : [25/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(13)\) | \(\approx\) | \(44.86300038\) |
| \(L(\frac12)\) | \(\approx\) | \(44.86300038\) |
| \(L(\frac{27}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( ( 1 + 2115 p T + 8643459 p^{3} T^{2} + 224791425 p^{10} T^{3} + 1293377472753 p^{11} T^{4} + 31213631650995 p^{18} T^{5} + 1410536132143689 p^{26} T^{6} + 31213631650995 p^{43} T^{7} + 1293377472753 p^{61} T^{8} + 224791425 p^{85} T^{9} + 8643459 p^{103} T^{10} + 2115 p^{126} T^{11} + p^{150} T^{12} )^{2} \) |
| 3 | \( 1 + 864933325756 p T^{2} + \)\(20\!\cdots\!70\)\( p^{5} T^{4} + \)\(13\!\cdots\!84\)\( p^{12} T^{6} + \)\(22\!\cdots\!23\)\( p^{18} T^{8} + \)\(96\!\cdots\!24\)\( p^{23} T^{10} + \)\(45\!\cdots\!48\)\( p^{32} T^{12} + \)\(96\!\cdots\!24\)\( p^{73} T^{14} + \)\(22\!\cdots\!23\)\( p^{118} T^{16} + \)\(13\!\cdots\!84\)\( p^{162} T^{18} + \)\(20\!\cdots\!70\)\( p^{205} T^{20} + 864933325756 p^{251} T^{22} + p^{300} T^{24} \) | |
| 5 | \( 1 + 639769845462814284 T^{2} + \)\(12\!\cdots\!46\)\( p^{2} T^{4} + \)\(68\!\cdots\!56\)\( p^{6} T^{6} + \)\(45\!\cdots\!11\)\( p^{10} T^{8} + \)\(23\!\cdots\!76\)\( p^{14} T^{10} + \)\(12\!\cdots\!36\)\( p^{18} T^{12} + \)\(23\!\cdots\!76\)\( p^{64} T^{14} + \)\(45\!\cdots\!11\)\( p^{110} T^{16} + \)\(68\!\cdots\!56\)\( p^{156} T^{18} + \)\(12\!\cdots\!46\)\( p^{202} T^{20} + 639769845462814284 p^{250} T^{22} + p^{300} T^{24} \) | |
| 11 | \( ( 1 - 19796727838824 T + \)\(44\!\cdots\!78\)\( p T^{2} - \)\(45\!\cdots\!72\)\( p^{2} T^{3} + \)\(57\!\cdots\!51\)\( p^{4} T^{4} - \)\(38\!\cdots\!00\)\( p^{6} T^{5} + \)\(43\!\cdots\!12\)\( p^{8} T^{6} - \)\(38\!\cdots\!00\)\( p^{31} T^{7} + \)\(57\!\cdots\!51\)\( p^{54} T^{8} - \)\(45\!\cdots\!72\)\( p^{77} T^{9} + \)\(44\!\cdots\!78\)\( p^{101} T^{10} - 19796727838824 p^{125} T^{11} + p^{150} T^{12} )^{2} \) | |
| 13 | \( 1 + \)\(27\!\cdots\!88\)\( T^{2} + \)\(43\!\cdots\!50\)\( T^{4} + \)\(44\!\cdots\!24\)\( T^{6} + \)\(26\!\cdots\!99\)\( p T^{8} + \)\(96\!\cdots\!24\)\( p^{3} T^{10} + \)\(37\!\cdots\!16\)\( p^{5} T^{12} + \)\(96\!\cdots\!24\)\( p^{53} T^{14} + \)\(26\!\cdots\!99\)\( p^{101} T^{16} + \)\(44\!\cdots\!24\)\( p^{150} T^{18} + \)\(43\!\cdots\!50\)\( p^{200} T^{20} + \)\(27\!\cdots\!88\)\( p^{250} T^{22} + p^{300} T^{24} \) | |
| 17 | \( 1 + \)\(15\!\cdots\!88\)\( p T^{2} + \)\(13\!\cdots\!50\)\( p^{2} T^{4} + \)\(89\!\cdots\!32\)\( p^{3} T^{6} + \)\(46\!\cdots\!27\)\( p^{4} T^{8} + \)\(20\!\cdots\!96\)\( p^{5} T^{10} + \)\(43\!\cdots\!68\)\( p^{7} T^{12} + \)\(20\!\cdots\!96\)\( p^{55} T^{14} + \)\(46\!\cdots\!27\)\( p^{104} T^{16} + \)\(89\!\cdots\!32\)\( p^{153} T^{18} + \)\(13\!\cdots\!50\)\( p^{202} T^{20} + \)\(15\!\cdots\!88\)\( p^{251} T^{22} + p^{300} T^{24} \) | |
| 19 | \( 1 + \)\(23\!\cdots\!76\)\( T^{2} + \)\(13\!\cdots\!38\)\( p T^{4} + \)\(46\!\cdots\!28\)\( p^{3} T^{6} + \)\(20\!\cdots\!57\)\( p^{5} T^{8} + \)\(55\!\cdots\!48\)\( p^{7} T^{10} + \)\(11\!\cdots\!92\)\( p^{9} T^{12} + \)\(55\!\cdots\!48\)\( p^{57} T^{14} + \)\(20\!\cdots\!57\)\( p^{105} T^{16} + \)\(46\!\cdots\!28\)\( p^{153} T^{18} + \)\(13\!\cdots\!38\)\( p^{201} T^{20} + \)\(23\!\cdots\!76\)\( p^{250} T^{22} + p^{300} T^{24} \) | |
| 23 | \( ( 1 - 1678195274964480 p T + \)\(55\!\cdots\!62\)\( p^{2} T^{2} - \)\(22\!\cdots\!00\)\( p^{3} T^{3} + \)\(15\!\cdots\!39\)\( p^{4} T^{4} - \)\(34\!\cdots\!80\)\( p^{5} T^{5} + \)\(37\!\cdots\!16\)\( p^{6} T^{6} - \)\(34\!\cdots\!80\)\( p^{30} T^{7} + \)\(15\!\cdots\!39\)\( p^{54} T^{8} - \)\(22\!\cdots\!00\)\( p^{78} T^{9} + \)\(55\!\cdots\!62\)\( p^{102} T^{10} - 1678195274964480 p^{126} T^{11} + p^{150} T^{12} )^{2} \) | |
| 29 | \( ( 1 - 4347249541361668044 T + \)\(15\!\cdots\!46\)\( T^{2} - \)\(41\!\cdots\!32\)\( T^{3} + \)\(10\!\cdots\!95\)\( T^{4} - \)\(21\!\cdots\!88\)\( T^{5} + \)\(43\!\cdots\!00\)\( T^{6} - \)\(21\!\cdots\!88\)\( p^{25} T^{7} + \)\(10\!\cdots\!95\)\( p^{50} T^{8} - \)\(41\!\cdots\!32\)\( p^{75} T^{9} + \)\(15\!\cdots\!46\)\( p^{100} T^{10} - 4347249541361668044 p^{125} T^{11} + p^{150} T^{12} )^{2} \) | |
| 31 | \( 1 + \)\(15\!\cdots\!80\)\( T^{2} + \)\(11\!\cdots\!02\)\( T^{4} + \)\(52\!\cdots\!28\)\( T^{6} + \)\(18\!\cdots\!23\)\( T^{8} + \)\(47\!\cdots\!08\)\( T^{10} + \)\(10\!\cdots\!80\)\( T^{12} + \)\(47\!\cdots\!08\)\( p^{50} T^{14} + \)\(18\!\cdots\!23\)\( p^{100} T^{16} + \)\(52\!\cdots\!28\)\( p^{150} T^{18} + \)\(11\!\cdots\!02\)\( p^{200} T^{20} + \)\(15\!\cdots\!80\)\( p^{250} T^{22} + p^{300} T^{24} \) | |
| 37 | \( ( 1 - 75926712500798424060 T + \)\(99\!\cdots\!46\)\( T^{2} - \)\(59\!\cdots\!40\)\( T^{3} + \)\(40\!\cdots\!47\)\( T^{4} - \)\(18\!\cdots\!40\)\( T^{5} + \)\(87\!\cdots\!36\)\( T^{6} - \)\(18\!\cdots\!40\)\( p^{25} T^{7} + \)\(40\!\cdots\!47\)\( p^{50} T^{8} - \)\(59\!\cdots\!40\)\( p^{75} T^{9} + \)\(99\!\cdots\!46\)\( p^{100} T^{10} - 75926712500798424060 p^{125} T^{11} + p^{150} T^{12} )^{2} \) | |
| 41 | \( 1 + \)\(82\!\cdots\!40\)\( T^{2} + \)\(11\!\cdots\!42\)\( T^{4} + \)\(19\!\cdots\!48\)\( T^{6} + \)\(20\!\cdots\!23\)\( p T^{8} + \)\(14\!\cdots\!48\)\( T^{10} + \)\(44\!\cdots\!00\)\( T^{12} + \)\(14\!\cdots\!48\)\( p^{50} T^{14} + \)\(20\!\cdots\!23\)\( p^{101} T^{16} + \)\(19\!\cdots\!48\)\( p^{150} T^{18} + \)\(11\!\cdots\!42\)\( p^{200} T^{20} + \)\(82\!\cdots\!40\)\( p^{250} T^{22} + p^{300} T^{24} \) | |
| 43 | \( ( 1 + \)\(19\!\cdots\!00\)\( T + \)\(22\!\cdots\!58\)\( T^{2} + \)\(18\!\cdots\!80\)\( T^{3} + \)\(25\!\cdots\!39\)\( T^{4} + \)\(18\!\cdots\!40\)\( T^{5} + \)\(22\!\cdots\!84\)\( T^{6} + \)\(18\!\cdots\!40\)\( p^{25} T^{7} + \)\(25\!\cdots\!39\)\( p^{50} T^{8} + \)\(18\!\cdots\!80\)\( p^{75} T^{9} + \)\(22\!\cdots\!58\)\( p^{100} T^{10} + \)\(19\!\cdots\!00\)\( p^{125} T^{11} + p^{150} T^{12} )^{2} \) | |
| 47 | \( 1 + \)\(45\!\cdots\!72\)\( T^{2} + \)\(10\!\cdots\!30\)\( T^{4} + \)\(16\!\cdots\!16\)\( T^{6} + \)\(18\!\cdots\!67\)\( T^{8} + \)\(16\!\cdots\!12\)\( T^{10} + \)\(11\!\cdots\!28\)\( T^{12} + \)\(16\!\cdots\!12\)\( p^{50} T^{14} + \)\(18\!\cdots\!67\)\( p^{100} T^{16} + \)\(16\!\cdots\!16\)\( p^{150} T^{18} + \)\(10\!\cdots\!30\)\( p^{200} T^{20} + \)\(45\!\cdots\!72\)\( p^{250} T^{22} + p^{300} T^{24} \) | |
| 53 | \( ( 1 - \)\(88\!\cdots\!00\)\( T + \)\(71\!\cdots\!58\)\( T^{2} - \)\(40\!\cdots\!00\)\( T^{3} + \)\(18\!\cdots\!35\)\( T^{4} - \)\(80\!\cdots\!00\)\( T^{5} + \)\(29\!\cdots\!40\)\( T^{6} - \)\(80\!\cdots\!00\)\( p^{25} T^{7} + \)\(18\!\cdots\!35\)\( p^{50} T^{8} - \)\(40\!\cdots\!00\)\( p^{75} T^{9} + \)\(71\!\cdots\!58\)\( p^{100} T^{10} - \)\(88\!\cdots\!00\)\( p^{125} T^{11} + p^{150} T^{12} )^{2} \) | |
| 59 | \( 1 + \)\(14\!\cdots\!16\)\( T^{2} + \)\(10\!\cdots\!82\)\( T^{4} + \)\(52\!\cdots\!72\)\( T^{6} + \)\(18\!\cdots\!23\)\( T^{8} + \)\(49\!\cdots\!12\)\( T^{10} + \)\(10\!\cdots\!88\)\( T^{12} + \)\(49\!\cdots\!12\)\( p^{50} T^{14} + \)\(18\!\cdots\!23\)\( p^{100} T^{16} + \)\(52\!\cdots\!72\)\( p^{150} T^{18} + \)\(10\!\cdots\!82\)\( p^{200} T^{20} + \)\(14\!\cdots\!16\)\( p^{250} T^{22} + p^{300} T^{24} \) | |
| 61 | \( 1 + \)\(30\!\cdots\!12\)\( T^{2} + \)\(41\!\cdots\!66\)\( T^{4} + \)\(35\!\cdots\!20\)\( T^{6} + \)\(22\!\cdots\!95\)\( T^{8} + \)\(12\!\cdots\!92\)\( T^{10} + \)\(57\!\cdots\!24\)\( T^{12} + \)\(12\!\cdots\!92\)\( p^{50} T^{14} + \)\(22\!\cdots\!95\)\( p^{100} T^{16} + \)\(35\!\cdots\!20\)\( p^{150} T^{18} + \)\(41\!\cdots\!66\)\( p^{200} T^{20} + \)\(30\!\cdots\!12\)\( p^{250} T^{22} + p^{300} T^{24} \) | |
| 67 | \( ( 1 - \)\(74\!\cdots\!80\)\( T + \)\(15\!\cdots\!30\)\( T^{2} - \)\(82\!\cdots\!80\)\( T^{3} + \)\(11\!\cdots\!35\)\( T^{4} - \)\(59\!\cdots\!20\)\( T^{5} + \)\(65\!\cdots\!36\)\( T^{6} - \)\(59\!\cdots\!20\)\( p^{25} T^{7} + \)\(11\!\cdots\!35\)\( p^{50} T^{8} - \)\(82\!\cdots\!80\)\( p^{75} T^{9} + \)\(15\!\cdots\!30\)\( p^{100} T^{10} - \)\(74\!\cdots\!80\)\( p^{125} T^{11} + p^{150} T^{12} )^{2} \) | |
| 71 | \( ( 1 - \)\(28\!\cdots\!92\)\( T + \)\(81\!\cdots\!94\)\( T^{2} - \)\(14\!\cdots\!84\)\( T^{3} + \)\(28\!\cdots\!15\)\( T^{4} - \)\(38\!\cdots\!24\)\( T^{5} + \)\(61\!\cdots\!80\)\( T^{6} - \)\(38\!\cdots\!24\)\( p^{25} T^{7} + \)\(28\!\cdots\!15\)\( p^{50} T^{8} - \)\(14\!\cdots\!84\)\( p^{75} T^{9} + \)\(81\!\cdots\!94\)\( p^{100} T^{10} - \)\(28\!\cdots\!92\)\( p^{125} T^{11} + p^{150} T^{12} )^{2} \) | |
| 73 | \( 1 + \)\(51\!\cdots\!04\)\( T^{2} + \)\(15\!\cdots\!70\)\( T^{4} + \)\(12\!\cdots\!44\)\( T^{6} + \)\(16\!\cdots\!47\)\( T^{8} - \)\(18\!\cdots\!32\)\( T^{10} + \)\(35\!\cdots\!64\)\( T^{12} - \)\(18\!\cdots\!32\)\( p^{50} T^{14} + \)\(16\!\cdots\!47\)\( p^{100} T^{16} + \)\(12\!\cdots\!44\)\( p^{150} T^{18} + \)\(15\!\cdots\!70\)\( p^{200} T^{20} + \)\(51\!\cdots\!04\)\( p^{250} T^{22} + p^{300} T^{24} \) | |
| 79 | \( ( 1 - \)\(74\!\cdots\!52\)\( T + \)\(10\!\cdots\!70\)\( T^{2} - \)\(67\!\cdots\!36\)\( T^{3} + \)\(63\!\cdots\!07\)\( T^{4} - \)\(31\!\cdots\!32\)\( T^{5} + \)\(21\!\cdots\!48\)\( T^{6} - \)\(31\!\cdots\!32\)\( p^{25} T^{7} + \)\(63\!\cdots\!07\)\( p^{50} T^{8} - \)\(67\!\cdots\!36\)\( p^{75} T^{9} + \)\(10\!\cdots\!70\)\( p^{100} T^{10} - \)\(74\!\cdots\!52\)\( p^{125} T^{11} + p^{150} T^{12} )^{2} \) | |
| 83 | \( 1 + \)\(28\!\cdots\!68\)\( T^{2} + \)\(43\!\cdots\!10\)\( T^{4} + \)\(53\!\cdots\!44\)\( T^{6} + \)\(58\!\cdots\!47\)\( T^{8} + \)\(63\!\cdots\!48\)\( T^{10} + \)\(65\!\cdots\!68\)\( T^{12} + \)\(63\!\cdots\!48\)\( p^{50} T^{14} + \)\(58\!\cdots\!47\)\( p^{100} T^{16} + \)\(53\!\cdots\!44\)\( p^{150} T^{18} + \)\(43\!\cdots\!10\)\( p^{200} T^{20} + \)\(28\!\cdots\!68\)\( p^{250} T^{22} + p^{300} T^{24} \) | |
| 89 | \( 1 + \)\(28\!\cdots\!20\)\( T^{2} + \)\(47\!\cdots\!02\)\( T^{4} + \)\(56\!\cdots\!72\)\( T^{6} + \)\(51\!\cdots\!23\)\( T^{8} + \)\(37\!\cdots\!92\)\( T^{10} + \)\(22\!\cdots\!80\)\( T^{12} + \)\(37\!\cdots\!92\)\( p^{50} T^{14} + \)\(51\!\cdots\!23\)\( p^{100} T^{16} + \)\(56\!\cdots\!72\)\( p^{150} T^{18} + \)\(47\!\cdots\!02\)\( p^{200} T^{20} + \)\(28\!\cdots\!20\)\( p^{250} T^{22} + p^{300} T^{24} \) | |
| 97 | \( 1 + \)\(41\!\cdots\!44\)\( T^{2} + \)\(82\!\cdots\!34\)\( T^{4} + \)\(10\!\cdots\!60\)\( T^{6} + \)\(96\!\cdots\!95\)\( T^{8} + \)\(66\!\cdots\!44\)\( T^{10} + \)\(35\!\cdots\!16\)\( T^{12} + \)\(66\!\cdots\!44\)\( p^{50} T^{14} + \)\(96\!\cdots\!95\)\( p^{100} T^{16} + \)\(10\!\cdots\!60\)\( p^{150} T^{18} + \)\(82\!\cdots\!34\)\( p^{200} T^{20} + \)\(41\!\cdots\!44\)\( p^{250} T^{22} + p^{300} T^{24} \) | |
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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
−2.34152193395742032414979315323, −2.33940974577916683264577619966, −2.29933586161611985967627526516, −2.22152589158804324214045879382, −2.18643534304302787127848313844, −1.79954910001715083379073885203, −1.74291549772279875746793006110, −1.52195421540971887158138478698, −1.51267991150392141665236958075, −1.49107331552511363092812636315, −1.39160946827161756876162835253, −1.20344823234336122353437087376, −1.12417805071902420000706925229, −1.09897542637285295566838796580, −0.899247983339637307509574767376, −0.845999361203040906433918386758, −0.76696753941859312778331653154, −0.63727493308412796638631847023, −0.62631776766334355763617260356, −0.48071225608662265303255322401, −0.39699950675252461574221265553, −0.38041916586219181512249081734, −0.32324714189595986379688132639, −0.29843803017951276834207425838, −0.29592754454222773772344477945, 0.29592754454222773772344477945, 0.29843803017951276834207425838, 0.32324714189595986379688132639, 0.38041916586219181512249081734, 0.39699950675252461574221265553, 0.48071225608662265303255322401, 0.62631776766334355763617260356, 0.63727493308412796638631847023, 0.76696753941859312778331653154, 0.845999361203040906433918386758, 0.899247983339637307509574767376, 1.09897542637285295566838796580, 1.12417805071902420000706925229, 1.20344823234336122353437087376, 1.39160946827161756876162835253, 1.49107331552511363092812636315, 1.51267991150392141665236958075, 1.52195421540971887158138478698, 1.74291549772279875746793006110, 1.79954910001715083379073885203, 2.18643534304302787127848313844, 2.22152589158804324214045879382, 2.29933586161611985967627526516, 2.33940974577916683264577619966, 2.34152193395742032414979315323
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.