Dirichlet series
| L(s) = 1 | − 6·2-s + 18·4-s − 8·7-s − 44·8-s − 13·9-s + 16·13-s + 48·14-s + 108·16-s + 40·17-s + 78·18-s − 26·19-s − 80·23-s − 27·25-s − 96·26-s − 144·28-s + 16·29-s − 32·31-s − 240·32-s − 240·34-s − 234·36-s − 148·37-s + 156·38-s + 66·41-s + 152·43-s + 480·46-s − 112·47-s + 99·49-s + ⋯ |
| L(s) = 1 | − 3·2-s + 9/2·4-s − 8/7·7-s − 5.5·8-s − 1.44·9-s + 1.23·13-s + 24/7·14-s + 27/4·16-s + 2.35·17-s + 13/3·18-s − 1.36·19-s − 3.47·23-s − 1.07·25-s − 3.69·26-s − 5.14·28-s + 0.551·29-s − 1.03·31-s − 7.5·32-s − 7.05·34-s − 6.5·36-s − 4·37-s + 4.10·38-s + 1.60·41-s + 3.53·43-s + 10.4·46-s − 2.38·47-s + 2.02·49-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s+1)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 37^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4516.49\) |
| Root analytic conductor: | \(1.41998\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 37^{12} ,\ ( \ : [1]^{12} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.1587476145\) |
| \(L(\frac12)\) | \(\approx\) | \(0.1587476145\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + p T + p T^{2} + p^{2} T^{3} + p^{2} T^{4} )^{3} \) |
| 37 | \( 1 + 4 p T + 10300 T^{2} + 399716 T^{3} + 130480 p T^{4} - 275180 p^{2} T^{5} - 470026 p^{3} T^{6} - 275180 p^{4} T^{7} + 130480 p^{5} T^{8} + 399716 p^{6} T^{9} + 10300 p^{8} T^{10} + 4 p^{11} T^{11} + p^{12} T^{12} \) | |
| good | 3 | \( 1 + 13 T^{2} + 2 p^{2} T^{4} - 40 p^{2} T^{5} - 1505 T^{6} - 1760 p T^{7} - 11372 T^{8} - 4760 p T^{9} + 34307 p T^{10} + 35800 p^{2} T^{11} + 259372 p^{2} T^{12} + 35800 p^{4} T^{13} + 34307 p^{5} T^{14} - 4760 p^{7} T^{15} - 11372 p^{8} T^{16} - 1760 p^{11} T^{17} - 1505 p^{12} T^{18} - 40 p^{16} T^{19} + 2 p^{18} T^{20} + 13 p^{20} T^{22} + p^{24} T^{24} \) |
| 5 | \( ( 1 + 9 T^{2} + 132 T^{3} - 316 T^{4} + 132 p^{2} T^{5} + 9 p^{4} T^{6} + p^{8} T^{8} )^{3} \) | |
| 7 | \( 1 + 8 T - 5 p T^{2} - 120 T^{3} - 514 T^{4} - 20360 T^{5} + 24901 p T^{6} + 729112 T^{7} - 5365000 T^{8} + 8445112 p T^{9} - 34494273 p T^{10} - 1485746040 T^{11} + 40626639204 T^{12} - 1485746040 p^{2} T^{13} - 34494273 p^{5} T^{14} + 8445112 p^{7} T^{15} - 5365000 p^{8} T^{16} + 729112 p^{10} T^{17} + 24901 p^{13} T^{18} - 20360 p^{14} T^{19} - 514 p^{16} T^{20} - 120 p^{18} T^{21} - 5 p^{21} T^{22} + 8 p^{22} T^{23} + p^{24} T^{24} \) | |
| 11 | \( 1 - 620 T^{2} + 202034 T^{4} - 46183740 T^{6} + 8484890383 T^{8} - 119941474760 p T^{10} + 173931051431740 T^{12} - 119941474760 p^{5} T^{14} + 8484890383 p^{8} T^{16} - 46183740 p^{12} T^{18} + 202034 p^{16} T^{20} - 620 p^{20} T^{22} + p^{24} T^{24} \) | |
| 13 | \( 1 - 16 T + 701 T^{2} - 8604 T^{3} + 259158 T^{4} - 3406444 T^{5} + 79809295 T^{6} - 1059125272 T^{7} + 19446438324 T^{8} - 249873128616 T^{9} + 4035863730881 T^{10} - 3910366940260 p T^{11} + 4399936607816 p^{2} T^{12} - 3910366940260 p^{3} T^{13} + 4035863730881 p^{4} T^{14} - 249873128616 p^{6} T^{15} + 19446438324 p^{8} T^{16} - 1059125272 p^{10} T^{17} + 79809295 p^{12} T^{18} - 3406444 p^{14} T^{19} + 259158 p^{16} T^{20} - 8604 p^{18} T^{21} + 701 p^{20} T^{22} - 16 p^{22} T^{23} + p^{24} T^{24} \) | |
| 17 | \( 1 - 40 T - 5 p T^{2} + 30004 T^{3} - 358973 T^{4} - 8622076 T^{5} + 193400728 T^{6} + 1410367700 T^{7} - 3303433323 p T^{8} - 339767350244 T^{9} + 18092527326557 T^{10} + 60719406954176 T^{11} - 6157572347446290 T^{12} + 60719406954176 p^{2} T^{13} + 18092527326557 p^{4} T^{14} - 339767350244 p^{6} T^{15} - 3303433323 p^{9} T^{16} + 1410367700 p^{10} T^{17} + 193400728 p^{12} T^{18} - 8622076 p^{14} T^{19} - 358973 p^{16} T^{20} + 30004 p^{18} T^{21} - 5 p^{21} T^{22} - 40 p^{22} T^{23} + p^{24} T^{24} \) | |
| 19 | \( 1 + 26 T + 23 p T^{2} - 3018 T^{3} - 210350 T^{4} - 4150238 T^{5} - 19929513 T^{6} + 902928062 T^{7} + 20399740880 T^{8} + 179669364642 T^{9} - 2946343505123 T^{10} - 65222176615634 T^{11} - 1207067327503748 T^{12} - 65222176615634 p^{2} T^{13} - 2946343505123 p^{4} T^{14} + 179669364642 p^{6} T^{15} + 20399740880 p^{8} T^{16} + 902928062 p^{10} T^{17} - 19929513 p^{12} T^{18} - 4150238 p^{14} T^{19} - 210350 p^{16} T^{20} - 3018 p^{18} T^{21} + 23 p^{21} T^{22} + 26 p^{22} T^{23} + p^{24} T^{24} \) | |
| 23 | \( 1 + 80 T + 3200 T^{2} + 84208 T^{3} + 2517154 T^{4} + 90235856 T^{5} + 2709469312 T^{6} + 59140951280 T^{7} + 1339695319359 T^{8} + 38873483371456 T^{9} + 1054098210602240 T^{10} + 20897058145001024 T^{11} + 413281343765872572 T^{12} + 20897058145001024 p^{2} T^{13} + 1054098210602240 p^{4} T^{14} + 38873483371456 p^{6} T^{15} + 1339695319359 p^{8} T^{16} + 59140951280 p^{10} T^{17} + 2709469312 p^{12} T^{18} + 90235856 p^{14} T^{19} + 2517154 p^{16} T^{20} + 84208 p^{18} T^{21} + 3200 p^{20} T^{22} + 80 p^{22} T^{23} + p^{24} T^{24} \) | |
| 29 | \( 1 - 16 T + 128 T^{2} + 42652 T^{3} - 1360928 T^{4} + 20328272 T^{5} + 758542984 T^{6} - 26919625936 T^{7} + 837544513056 T^{8} + 10812258805876 T^{9} - 189436016605312 T^{10} + 6657402982055312 T^{11} + 135620371555263918 T^{12} + 6657402982055312 p^{2} T^{13} - 189436016605312 p^{4} T^{14} + 10812258805876 p^{6} T^{15} + 837544513056 p^{8} T^{16} - 26919625936 p^{10} T^{17} + 758542984 p^{12} T^{18} + 20328272 p^{14} T^{19} - 1360928 p^{16} T^{20} + 42652 p^{18} T^{21} + 128 p^{20} T^{22} - 16 p^{22} T^{23} + p^{24} T^{24} \) | |
| 31 | \( 1 + 32 T + 512 T^{2} + 44160 T^{3} + 1706658 T^{4} - 4031200 T^{5} - 27754496 T^{6} + 5101075712 T^{7} - 248118458241 T^{8} - 20835987142944 T^{9} - 258958715158528 T^{10} - 9487825155475360 T^{11} + 118113147956269244 T^{12} - 9487825155475360 p^{2} T^{13} - 258958715158528 p^{4} T^{14} - 20835987142944 p^{6} T^{15} - 248118458241 p^{8} T^{16} + 5101075712 p^{10} T^{17} - 27754496 p^{12} T^{18} - 4031200 p^{14} T^{19} + 1706658 p^{16} T^{20} + 44160 p^{18} T^{21} + 512 p^{20} T^{22} + 32 p^{22} T^{23} + p^{24} T^{24} \) | |
| 41 | \( 1 - 66 T + 9223 T^{2} - 512886 T^{3} + 41741975 T^{4} - 2153599560 T^{5} + 138039902004 T^{6} - 6652438106940 T^{7} + 369042941916277 T^{8} - 16251059475428514 T^{9} + 808934478208152869 T^{10} - 32579379102876646434 T^{11} + \)\(14\!\cdots\!46\)\( T^{12} - 32579379102876646434 p^{2} T^{13} + 808934478208152869 p^{4} T^{14} - 16251059475428514 p^{6} T^{15} + 369042941916277 p^{8} T^{16} - 6652438106940 p^{10} T^{17} + 138039902004 p^{12} T^{18} - 2153599560 p^{14} T^{19} + 41741975 p^{16} T^{20} - 512886 p^{18} T^{21} + 9223 p^{20} T^{22} - 66 p^{22} T^{23} + p^{24} T^{24} \) | |
| 43 | \( 1 - 152 T + 11552 T^{2} - 645480 T^{3} + 27290994 T^{4} - 850122056 T^{5} + 22275205024 T^{6} - 451362708152 T^{7} + 16542428878815 T^{8} - 2249703667494000 T^{9} + 195643688299049792 T^{10} - 12718047251750264336 T^{11} + \)\(64\!\cdots\!80\)\( T^{12} - 12718047251750264336 p^{2} T^{13} + 195643688299049792 p^{4} T^{14} - 2249703667494000 p^{6} T^{15} + 16542428878815 p^{8} T^{16} - 451362708152 p^{10} T^{17} + 22275205024 p^{12} T^{18} - 850122056 p^{14} T^{19} + 27290994 p^{16} T^{20} - 645480 p^{18} T^{21} + 11552 p^{20} T^{22} - 152 p^{22} T^{23} + p^{24} T^{24} \) | |
| 47 | \( ( 1 + 56 T + 11742 T^{2} + 541512 T^{3} + 1267673 p T^{4} + 2235474080 T^{5} + 170295528148 T^{6} + 2235474080 p^{2} T^{7} + 1267673 p^{5} T^{8} + 541512 p^{6} T^{9} + 11742 p^{8} T^{10} + 56 p^{10} T^{11} + p^{12} T^{12} )^{2} \) | |
| 53 | \( 1 - 74 T - 6791 T^{2} + 592326 T^{3} + 22400546 T^{4} - 2086928302 T^{5} - 85449307553 T^{6} + 5473753030010 T^{7} + 349541708987000 T^{8} - 14332892834483818 T^{9} - 18134152089947103 p T^{10} + 19387189891261941174 T^{11} + \)\(22\!\cdots\!00\)\( T^{12} + 19387189891261941174 p^{2} T^{13} - 18134152089947103 p^{5} T^{14} - 14332892834483818 p^{6} T^{15} + 349541708987000 p^{8} T^{16} + 5473753030010 p^{10} T^{17} - 85449307553 p^{12} T^{18} - 2086928302 p^{14} T^{19} + 22400546 p^{16} T^{20} + 592326 p^{18} T^{21} - 6791 p^{20} T^{22} - 74 p^{22} T^{23} + p^{24} T^{24} \) | |
| 59 | \( 1 + 114 T + 20421 T^{2} + 1360254 T^{3} + 151196286 T^{4} + 6662092482 T^{5} + 696707254467 T^{6} + 23867436846582 T^{7} + 3039925397464236 T^{8} + 108581868479120754 T^{9} + 14088740476658480901 T^{10} + \)\(52\!\cdots\!14\)\( T^{11} + \)\(56\!\cdots\!32\)\( T^{12} + \)\(52\!\cdots\!14\)\( p^{2} T^{13} + 14088740476658480901 p^{4} T^{14} + 108581868479120754 p^{6} T^{15} + 3039925397464236 p^{8} T^{16} + 23867436846582 p^{10} T^{17} + 696707254467 p^{12} T^{18} + 6662092482 p^{14} T^{19} + 151196286 p^{16} T^{20} + 1360254 p^{18} T^{21} + 20421 p^{20} T^{22} + 114 p^{22} T^{23} + p^{24} T^{24} \) | |
| 61 | \( 1 - 448 T + 96011 T^{2} - 12710700 T^{3} + 1107911571 T^{4} - 58820637316 T^{5} + 673977876232 T^{6} + 203125568801756 T^{7} - 21575121556392171 T^{8} + 1059658821026884668 T^{9} - 2656534997307293779 T^{10} - \)\(41\!\cdots\!32\)\( T^{11} + \)\(36\!\cdots\!70\)\( T^{12} - \)\(41\!\cdots\!32\)\( p^{2} T^{13} - 2656534997307293779 p^{4} T^{14} + 1059658821026884668 p^{6} T^{15} - 21575121556392171 p^{8} T^{16} + 203125568801756 p^{10} T^{17} + 673977876232 p^{12} T^{18} - 58820637316 p^{14} T^{19} + 1107911571 p^{16} T^{20} - 12710700 p^{18} T^{21} + 96011 p^{20} T^{22} - 448 p^{22} T^{23} + p^{24} T^{24} \) | |
| 67 | \( 1 - 468 T + 127649 T^{2} - 25571988 T^{3} + 4148829362 T^{4} - 571095004356 T^{5} + 68758378041315 T^{6} - 7380953256327084 T^{7} + 716360253453204700 T^{8} - 63460014861213596988 T^{9} + \)\(51\!\cdots\!45\)\( T^{10} - \)\(38\!\cdots\!12\)\( T^{11} + \)\(27\!\cdots\!20\)\( T^{12} - \)\(38\!\cdots\!12\)\( p^{2} T^{13} + \)\(51\!\cdots\!45\)\( p^{4} T^{14} - 63460014861213596988 p^{6} T^{15} + 716360253453204700 p^{8} T^{16} - 7380953256327084 p^{10} T^{17} + 68758378041315 p^{12} T^{18} - 571095004356 p^{14} T^{19} + 4148829362 p^{16} T^{20} - 25571988 p^{18} T^{21} + 127649 p^{20} T^{22} - 468 p^{22} T^{23} + p^{24} T^{24} \) | |
| 71 | \( 1 - 116 T - 8543 T^{2} + 1128780 T^{3} + 55427870 T^{4} - 4835517532 T^{5} - 405799335449 T^{6} + 8809907126516 T^{7} + 2736287971337312 T^{8} - 37439442662421076 T^{9} - 8931007726306710819 T^{10} + \)\(10\!\cdots\!72\)\( T^{11} + \)\(23\!\cdots\!88\)\( T^{12} + \)\(10\!\cdots\!72\)\( p^{2} T^{13} - 8931007726306710819 p^{4} T^{14} - 37439442662421076 p^{6} T^{15} + 2736287971337312 p^{8} T^{16} + 8809907126516 p^{10} T^{17} - 405799335449 p^{12} T^{18} - 4835517532 p^{14} T^{19} + 55427870 p^{16} T^{20} + 1128780 p^{18} T^{21} - 8543 p^{20} T^{22} - 116 p^{22} T^{23} + p^{24} T^{24} \) | |
| 73 | \( 1 - 29844 T^{2} + 466476354 T^{4} - 5046296641988 T^{6} + 42794129183463855 T^{8} - \)\(29\!\cdots\!04\)\( T^{10} + \)\(17\!\cdots\!04\)\( T^{12} - \)\(29\!\cdots\!04\)\( p^{4} T^{14} + 42794129183463855 p^{8} T^{16} - 5046296641988 p^{12} T^{18} + 466476354 p^{16} T^{20} - 29844 p^{20} T^{22} + p^{24} T^{24} \) | |
| 79 | \( 1 - 114 T + 27573 T^{2} - 2748374 T^{3} + 380581290 T^{4} - 28933803306 T^{5} + 2958679076927 T^{6} - 129382607548566 T^{7} + 9158084553476520 T^{8} + 501426613885122126 T^{9} - 53531739947560947747 T^{10} + \)\(11\!\cdots\!06\)\( T^{11} - \)\(69\!\cdots\!48\)\( T^{12} + \)\(11\!\cdots\!06\)\( p^{2} T^{13} - 53531739947560947747 p^{4} T^{14} + 501426613885122126 p^{6} T^{15} + 9158084553476520 p^{8} T^{16} - 129382607548566 p^{10} T^{17} + 2958679076927 p^{12} T^{18} - 28933803306 p^{14} T^{19} + 380581290 p^{16} T^{20} - 2748374 p^{18} T^{21} + 27573 p^{20} T^{22} - 114 p^{22} T^{23} + p^{24} T^{24} \) | |
| 83 | \( 1 + 20 T - 21527 T^{2} - 1168164 T^{3} + 187890890 T^{4} + 15412871476 T^{5} - 814495641965 T^{6} - 57649963055492 T^{7} + 6327346424906084 T^{8} - 312863629933056620 T^{9} - \)\(11\!\cdots\!51\)\( T^{10} + \)\(20\!\cdots\!84\)\( T^{11} + \)\(10\!\cdots\!12\)\( T^{12} + \)\(20\!\cdots\!84\)\( p^{2} T^{13} - \)\(11\!\cdots\!51\)\( p^{4} T^{14} - 312863629933056620 p^{6} T^{15} + 6327346424906084 p^{8} T^{16} - 57649963055492 p^{10} T^{17} - 814495641965 p^{12} T^{18} + 15412871476 p^{14} T^{19} + 187890890 p^{16} T^{20} - 1168164 p^{18} T^{21} - 21527 p^{20} T^{22} + 20 p^{22} T^{23} + p^{24} T^{24} \) | |
| 89 | \( 1 - 340 T + 56567 T^{2} - 4373600 T^{3} - 126304253 T^{4} + 71216046260 T^{5} - 7469032569872 T^{6} + 137079933328940 T^{7} + 57798155432720925 T^{8} - 7398743630294760560 T^{9} + \)\(23\!\cdots\!77\)\( T^{10} + \)\(42\!\cdots\!20\)\( T^{11} - \)\(65\!\cdots\!26\)\( T^{12} + \)\(42\!\cdots\!20\)\( p^{2} T^{13} + \)\(23\!\cdots\!77\)\( p^{4} T^{14} - 7398743630294760560 p^{6} T^{15} + 57798155432720925 p^{8} T^{16} + 137079933328940 p^{10} T^{17} - 7469032569872 p^{12} T^{18} + 71216046260 p^{14} T^{19} - 126304253 p^{16} T^{20} - 4373600 p^{18} T^{21} + 56567 p^{20} T^{22} - 340 p^{22} T^{23} + p^{24} T^{24} \) | |
| 97 | \( 1 + 356 T + 63368 T^{2} + 10498632 T^{3} + 1700796144 T^{4} + 205355771636 T^{5} + 20441241585136 T^{6} + 1999622271421244 T^{7} + 145029519802550352 T^{8} + 4635182012277166872 T^{9} - \)\(22\!\cdots\!32\)\( T^{10} - \)\(78\!\cdots\!48\)\( T^{11} - \)\(11\!\cdots\!30\)\( T^{12} - \)\(78\!\cdots\!48\)\( p^{2} T^{13} - \)\(22\!\cdots\!32\)\( p^{4} T^{14} + 4635182012277166872 p^{6} T^{15} + 145029519802550352 p^{8} T^{16} + 1999622271421244 p^{10} T^{17} + 20441241585136 p^{12} T^{18} + 205355771636 p^{14} T^{19} + 1700796144 p^{16} T^{20} + 10498632 p^{18} T^{21} + 63368 p^{20} T^{22} + 356 p^{22} T^{23} + p^{24} T^{24} \) | |
| 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
−5.19910654086863044525775874156, −5.06936399593585050851337281386, −5.04819526633100799745261451548, −5.00325734006819637408779381647, −4.97543934944028469957670803969, −4.08321219803295622830328155643, −3.91834189605795842790369683857, −3.89603598002285038353017031363, −3.88304832765194202988979697518, −3.87986094719531494706329588363, −3.81811121908059596922015516862, −3.66484966059280135599048120159, −3.57959017519975297459445528918, −3.11398359967208730135267676266, −3.01204521674564746820762768910, −2.49567421766388986815349330912, −2.48773111805036411749449610908, −2.45408804640071771921704156048, −2.22661997427279872365956695391, −2.18593451888178028677119272440, −1.73936633839074189077699221503, −1.22144517551066268921804337439, −1.05656480788068467339950249167, −0.56107691336599234447883931808, −0.29678572219178822443850836618, 0.29678572219178822443850836618, 0.56107691336599234447883931808, 1.05656480788068467339950249167, 1.22144517551066268921804337439, 1.73936633839074189077699221503, 2.18593451888178028677119272440, 2.22661997427279872365956695391, 2.45408804640071771921704156048, 2.48773111805036411749449610908, 2.49567421766388986815349330912, 3.01204521674564746820762768910, 3.11398359967208730135267676266, 3.57959017519975297459445528918, 3.66484966059280135599048120159, 3.81811121908059596922015516862, 3.87986094719531494706329588363, 3.88304832765194202988979697518, 3.89603598002285038353017031363, 3.91834189605795842790369683857, 4.08321219803295622830328155643, 4.97543934944028469957670803969, 5.00325734006819637408779381647, 5.04819526633100799745261451548, 5.06936399593585050851337281386, 5.19910654086863044525775874156
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.