Dirichlet series
| L(s) = 1 | + 120·7-s + 4.86e3·13-s + 6.57e3·16-s + 9.96e4·31-s + 1.41e5·37-s − 3.95e5·43-s + 7.20e3·49-s + 1.56e6·61-s + 2.29e6·67-s − 1.46e6·73-s + 5.83e5·91-s + 8.03e6·97-s − 5.39e6·103-s + 7.89e5·112-s − 8.80e6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.18e7·169-s + 173-s + 179-s + ⋯ |
| L(s) = 1 | + 0.349·7-s + 2.21·13-s + 1.60·16-s + 3.34·31-s + 2.79·37-s − 4.96·43-s + 0.0611·49-s + 6.90·61-s + 7.63·67-s − 3.77·73-s + 0.773·91-s + 8.80·97-s − 4.93·103-s + 0.561·112-s − 4.97·121-s + 2.44·169-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(7-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 5^{24}\right)^{s/2} \, \Gamma_{\C}(s+3)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{24} \cdot 5^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.69955\times 10^{20}\) |
| Root analytic conductor: | \(7.19459\) |
| Motivic weight: | \(6\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{24} \cdot 5^{24} ,\ ( \ : [3]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{7}{2})\) | \(\approx\) | \(6.417013440\) |
| \(L(\frac12)\) | \(\approx\) | \(6.417013440\) |
| \(L(4)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) | |
| good | 2 | \( 1 - 6579 T^{4} + 1128105 p^{2} T^{8} + 4876735 p^{14} T^{12} + 1128105 p^{26} T^{16} - 6579 p^{48} T^{20} + p^{72} T^{24} \) |
| 7 | \( ( 1 - 60 T + 1800 T^{2} + 72722060 T^{3} - 270669153 p^{2} T^{4} - 834431102160 p T^{5} + 3018583087523600 T^{6} - 834431102160 p^{7} T^{7} - 270669153 p^{14} T^{8} + 72722060 p^{18} T^{9} + 1800 p^{24} T^{10} - 60 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 11 | \( ( 1 + 4402866 T^{2} + 12149946447315 T^{4} + 26401283481062915620 T^{6} + 12149946447315 p^{12} T^{8} + 4402866 p^{24} T^{10} + p^{36} T^{12} )^{2} \) | |
| 13 | \( ( 1 - 2430 T + 2952450 T^{2} - 917887990 p T^{3} - 8203166214957 T^{4} + 78320049617720340 T^{5} - 94905480874973343100 T^{6} + 78320049617720340 p^{6} T^{7} - 8203166214957 p^{12} T^{8} - 917887990 p^{19} T^{9} + 2952450 p^{24} T^{10} - 2430 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 17 | \( 1 - 401614849782654 T^{4} + \)\(88\!\cdots\!35\)\( T^{8} - \)\(24\!\cdots\!00\)\( T^{12} + \)\(88\!\cdots\!35\)\( p^{24} T^{16} - 401614849782654 p^{48} T^{20} + p^{72} T^{24} \) | |
| 19 | \( ( 1 - 102373134 T^{2} + 3672062943779535 T^{4} - \)\(82\!\cdots\!00\)\( T^{6} + 3672062943779535 p^{12} T^{8} - 102373134 p^{24} T^{10} + p^{36} T^{12} )^{2} \) | |
| 23 | \( 1 + 2615710555531206 T^{4} + \)\(44\!\cdots\!35\)\( T^{8} - \)\(82\!\cdots\!00\)\( T^{12} + \)\(44\!\cdots\!35\)\( p^{24} T^{16} + 2615710555531206 p^{48} T^{20} + p^{72} T^{24} \) | |
| 29 | \( ( 1 - 2354698926 T^{2} + 2587431289287404115 T^{4} - \)\(18\!\cdots\!20\)\( T^{6} + 2587431289287404115 p^{12} T^{8} - 2354698926 p^{24} T^{10} + p^{36} T^{12} )^{2} \) | |
| 31 | \( ( 1 - 804 p T + 1453360035 T^{2} - 20833575087800 T^{3} + 1453360035 p^{6} T^{4} - 804 p^{13} T^{5} + p^{18} T^{6} )^{4} \) | |
| 37 | \( ( 1 - 70830 T + 2508444450 T^{2} - 195389684927470 T^{3} + 14025764450458291443 T^{4} - \)\(43\!\cdots\!60\)\( T^{5} + \)\(14\!\cdots\!00\)\( T^{6} - \)\(43\!\cdots\!60\)\( p^{6} T^{7} + 14025764450458291443 p^{12} T^{8} - 195389684927470 p^{18} T^{9} + 2508444450 p^{24} T^{10} - 70830 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 41 | \( ( 1 - 6741183054 T^{2} + 48309648508083397215 T^{4} - \)\(15\!\cdots\!80\)\( T^{6} + 48309648508083397215 p^{12} T^{8} - 6741183054 p^{24} T^{10} + p^{36} T^{12} )^{2} \) | |
| 43 | \( ( 1 + 197520 T + 19507075200 T^{2} + 1633149371270480 T^{3} + \)\(17\!\cdots\!03\)\( T^{4} + \)\(18\!\cdots\!40\)\( T^{5} + \)\(15\!\cdots\!00\)\( T^{6} + \)\(18\!\cdots\!40\)\( p^{6} T^{7} + \)\(17\!\cdots\!03\)\( p^{12} T^{8} + 1633149371270480 p^{18} T^{9} + 19507075200 p^{24} T^{10} + 197520 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 47 | \( 1 + \)\(23\!\cdots\!66\)\( T^{4} + \)\(37\!\cdots\!95\)\( T^{8} + \)\(56\!\cdots\!40\)\( T^{12} + \)\(37\!\cdots\!95\)\( p^{24} T^{16} + \)\(23\!\cdots\!66\)\( p^{48} T^{20} + p^{72} T^{24} \) | |
| 53 | \( 1 - \)\(24\!\cdots\!34\)\( T^{4} - \)\(13\!\cdots\!05\)\( T^{8} - \)\(18\!\cdots\!60\)\( T^{12} - \)\(13\!\cdots\!05\)\( p^{24} T^{16} - \)\(24\!\cdots\!34\)\( p^{48} T^{20} + p^{72} T^{24} \) | |
| 59 | \( ( 1 - 178482983346 T^{2} + \)\(15\!\cdots\!15\)\( T^{4} - \)\(82\!\cdots\!20\)\( T^{6} + \)\(15\!\cdots\!15\)\( p^{12} T^{8} - 178482983346 p^{24} T^{10} + p^{36} T^{12} )^{2} \) | |
| 61 | \( ( 1 - 391956 T + 188962613295 T^{2} - 41218053410672840 T^{3} + 188962613295 p^{6} T^{4} - 391956 p^{12} T^{5} + p^{18} T^{6} )^{4} \) | |
| 67 | \( ( 1 - 1148160 T + 659135692800 T^{2} - 278776770001263040 T^{3} + \)\(94\!\cdots\!83\)\( T^{4} - \)\(26\!\cdots\!20\)\( T^{5} + \)\(77\!\cdots\!00\)\( T^{6} - \)\(26\!\cdots\!20\)\( p^{6} T^{7} + \)\(94\!\cdots\!83\)\( p^{12} T^{8} - 278776770001263040 p^{18} T^{9} + 659135692800 p^{24} T^{10} - 1148160 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 71 | \( ( 1 + 474507959526 T^{2} + \)\(11\!\cdots\!15\)\( T^{4} + \)\(18\!\cdots\!20\)\( T^{6} + \)\(11\!\cdots\!15\)\( p^{12} T^{8} + 474507959526 p^{24} T^{10} + p^{36} T^{12} )^{2} \) | |
| 73 | \( ( 1 + 734010 T + 269385340050 T^{2} + 111734130849312890 T^{3} - \)\(21\!\cdots\!37\)\( T^{4} - \)\(32\!\cdots\!80\)\( T^{5} - \)\(11\!\cdots\!00\)\( T^{6} - \)\(32\!\cdots\!80\)\( p^{6} T^{7} - \)\(21\!\cdots\!37\)\( p^{12} T^{8} + 111734130849312890 p^{18} T^{9} + 269385340050 p^{24} T^{10} + 734010 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
| 79 | \( ( 1 - 984783846294 T^{2} + \)\(43\!\cdots\!35\)\( T^{4} - \)\(12\!\cdots\!00\)\( T^{6} + \)\(43\!\cdots\!35\)\( p^{12} T^{8} - 984783846294 p^{24} T^{10} + p^{36} T^{12} )^{2} \) | |
| 83 | \( 1 + \)\(59\!\cdots\!86\)\( T^{4} - \)\(49\!\cdots\!05\)\( T^{8} - \)\(23\!\cdots\!60\)\( T^{12} - \)\(49\!\cdots\!05\)\( p^{24} T^{16} + \)\(59\!\cdots\!86\)\( p^{48} T^{20} + p^{72} T^{24} \) | |
| 89 | \( ( 1 - 1637648083266 T^{2} + \)\(14\!\cdots\!15\)\( T^{4} - \)\(84\!\cdots\!20\)\( T^{6} + \)\(14\!\cdots\!15\)\( p^{12} T^{8} - 1637648083266 p^{24} T^{10} + p^{36} T^{12} )^{2} \) | |
| 97 | \( ( 1 - 4017510 T + 8070193300050 T^{2} - 12466768264810470790 T^{3} + \)\(16\!\cdots\!23\)\( T^{4} - \)\(18\!\cdots\!20\)\( T^{5} + \)\(17\!\cdots\!00\)\( T^{6} - \)\(18\!\cdots\!20\)\( p^{6} T^{7} + \)\(16\!\cdots\!23\)\( p^{12} T^{8} - 12466768264810470790 p^{18} T^{9} + 8070193300050 p^{24} T^{10} - 4017510 p^{30} T^{11} + p^{36} T^{12} )^{2} \) | |
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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
−3.20089620623078692521813233931, −3.03434614345971306658178315154, −2.65834415657096312390229910137, −2.64168158488900787514901995607, −2.45496361811491591118280865071, −2.45362323986800367727642636929, −2.31222364379714115039238550371, −2.25537884529000476458354626161, −2.19658035987045059067744423263, −2.08747260739425932266581863728, −2.06568213716496830171810095342, −1.62529955612288763273044460230, −1.57782170317735161022005979859, −1.31933667835530155738092792553, −1.14786077684938519926307655699, −1.13416270932839367527987832159, −1.10408586022333989533961525108, −1.04969810652908993174932377894, −0.994271797568139959802073427146, −0.928995846533963652549889803294, −0.789151553471750075502258707322, −0.38222503753523920725629378187, −0.37127942774889828030658664472, −0.23699398767659863592347014372, −0.07516370169444207601159972828, 0.07516370169444207601159972828, 0.23699398767659863592347014372, 0.37127942774889828030658664472, 0.38222503753523920725629378187, 0.789151553471750075502258707322, 0.928995846533963652549889803294, 0.994271797568139959802073427146, 1.04969810652908993174932377894, 1.10408586022333989533961525108, 1.13416270932839367527987832159, 1.14786077684938519926307655699, 1.31933667835530155738092792553, 1.57782170317735161022005979859, 1.62529955612288763273044460230, 2.06568213716496830171810095342, 2.08747260739425932266581863728, 2.19658035987045059067744423263, 2.25537884529000476458354626161, 2.31222364379714115039238550371, 2.45362323986800367727642636929, 2.45496361811491591118280865071, 2.64168158488900787514901995607, 2.65834415657096312390229910137, 3.03434614345971306658178315154, 3.20089620623078692521813233931
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.