Dirichlet series
| L(s) = 1 | + 2·2-s + 2·3-s + 2·4-s − 4·5-s + 4·6-s + 2·9-s − 8·10-s − 4·11-s + 4·12-s + 18·13-s − 8·15-s − 8·17-s + 4·18-s − 8·19-s − 8·20-s − 8·22-s + 8·25-s + 36·26-s + 8·27-s + 2·29-s − 16·30-s + 20·31-s − 8·33-s − 16·34-s + 4·36-s − 4·37-s − 16·38-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 4-s − 1.78·5-s + 1.63·6-s + 2/3·9-s − 2.52·10-s − 1.20·11-s + 1.15·12-s + 4.99·13-s − 2.06·15-s − 1.94·17-s + 0.942·18-s − 1.83·19-s − 1.78·20-s − 1.70·22-s + 8/5·25-s + 7.06·26-s + 1.53·27-s + 0.371·29-s − 2.92·30-s + 3.59·31-s − 1.39·33-s − 2.74·34-s + 2/3·36-s − 0.657·37-s − 2.59·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 23^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 23^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(414475.\) |
| Root analytic conductor: | \(1.71420\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{48} \cdot 23^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(22.42409851\) |
| \(L(\frac12)\) | \(\approx\) | \(22.42409851\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 - p T + p T^{2} - p^{2} T^{4} + p^{3} T^{5} - 3 p^{2} T^{6} + p^{4} T^{7} - p^{4} T^{8} + p^{5} T^{10} - p^{6} T^{11} + p^{6} T^{12} \) |
| 23 | \( ( 1 + T^{2} )^{6} \) | |
| good | 3 | \( 1 - 2 T + 2 T^{2} - 8 T^{3} + 5 T^{4} + 20 T^{5} - 2 p^{2} T^{6} + 70 T^{7} - 190 T^{8} + 2 p^{3} T^{9} + 2 T^{10} - 176 T^{11} + 1273 T^{12} - 176 p T^{13} + 2 p^{2} T^{14} + 2 p^{6} T^{15} - 190 p^{4} T^{16} + 70 p^{5} T^{17} - 2 p^{8} T^{18} + 20 p^{7} T^{19} + 5 p^{8} T^{20} - 8 p^{9} T^{21} + 2 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 + 4 T + 8 T^{2} - 78 T^{4} - 212 T^{5} - 224 T^{6} + 352 T^{7} + 2731 T^{8} + 4604 T^{9} + 1568 T^{10} - 17028 T^{11} - 69976 T^{12} - 17028 p T^{13} + 1568 p^{2} T^{14} + 4604 p^{3} T^{15} + 2731 p^{4} T^{16} + 352 p^{5} T^{17} - 224 p^{6} T^{18} - 212 p^{7} T^{19} - 78 p^{8} T^{20} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 - 68 T^{2} + 2206 T^{4} - 45092 T^{6} + 13191 p^{2} T^{8} - 977464 p T^{10} + 54838648 T^{12} - 977464 p^{3} T^{14} + 13191 p^{6} T^{16} - 45092 p^{6} T^{18} + 2206 p^{8} T^{20} - 68 p^{10} T^{22} + p^{12} T^{24} \) | |
| 11 | \( 1 + 4 T + 8 T^{2} + 52 T^{3} - 134 T^{4} - 1140 T^{5} - 2136 T^{6} - 13484 T^{7} - 22433 T^{8} + 75376 T^{9} + 143376 T^{10} + 912056 T^{11} + 5810608 T^{12} + 912056 p T^{13} + 143376 p^{2} T^{14} + 75376 p^{3} T^{15} - 22433 p^{4} T^{16} - 13484 p^{5} T^{17} - 2136 p^{6} T^{18} - 1140 p^{7} T^{19} - 134 p^{8} T^{20} + 52 p^{9} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 - 18 T + 162 T^{2} - 84 p T^{3} + 6673 T^{4} - 37448 T^{5} + 189270 T^{6} - 889826 T^{7} + 3998178 T^{8} - 17024206 T^{9} + 68598506 T^{10} - 265184208 T^{11} + 981270081 T^{12} - 265184208 p T^{13} + 68598506 p^{2} T^{14} - 17024206 p^{3} T^{15} + 3998178 p^{4} T^{16} - 889826 p^{5} T^{17} + 189270 p^{6} T^{18} - 37448 p^{7} T^{19} + 6673 p^{8} T^{20} - 84 p^{10} T^{21} + 162 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( ( 1 + 4 T + 64 T^{2} + 246 T^{3} + 2065 T^{4} + 430 p T^{5} + 43246 T^{6} + 430 p^{2} T^{7} + 2065 p^{2} T^{8} + 246 p^{3} T^{9} + 64 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( 1 + 8 T + 32 T^{2} + 8 p T^{3} + 1418 T^{4} + 8680 T^{5} + 35616 T^{6} + 179896 T^{7} + 1102335 T^{8} + 5132496 T^{9} + 20297152 T^{10} + 101957616 T^{11} + 511158412 T^{12} + 101957616 p T^{13} + 20297152 p^{2} T^{14} + 5132496 p^{3} T^{15} + 1102335 p^{4} T^{16} + 179896 p^{5} T^{17} + 35616 p^{6} T^{18} + 8680 p^{7} T^{19} + 1418 p^{8} T^{20} + 8 p^{10} T^{21} + 32 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 2 T + 2 T^{2} - 424 T^{3} + 2329 T^{4} + 1780 T^{5} + 81670 T^{6} - 841602 T^{7} + 1013250 T^{8} - 114454 p T^{9} + 172825850 T^{10} - 584430420 T^{11} - 337880919 T^{12} - 584430420 p T^{13} + 172825850 p^{2} T^{14} - 114454 p^{4} T^{15} + 1013250 p^{4} T^{16} - 841602 p^{5} T^{17} + 81670 p^{6} T^{18} + 1780 p^{7} T^{19} + 2329 p^{8} T^{20} - 424 p^{9} T^{21} + 2 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( ( 1 - 10 T + 143 T^{2} - 1220 T^{3} + 9818 T^{4} - 67262 T^{5} + 389725 T^{6} - 67262 p T^{7} + 9818 p^{2} T^{8} - 1220 p^{3} T^{9} + 143 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 37 | \( 1 + 4 T + 8 T^{2} + 232 T^{3} + 14 p T^{4} - 9756 T^{5} - 16256 T^{6} - 371240 T^{7} - 61273 p T^{8} + 5079796 T^{9} + 8338688 T^{10} + 230905436 T^{11} + 6813255824 T^{12} + 230905436 p T^{13} + 8338688 p^{2} T^{14} + 5079796 p^{3} T^{15} - 61273 p^{5} T^{16} - 371240 p^{5} T^{17} - 16256 p^{6} T^{18} - 9756 p^{7} T^{19} + 14 p^{9} T^{20} + 232 p^{9} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 254 T^{2} + 32757 T^{4} - 2923858 T^{6} + 201141458 T^{8} - 11090716582 T^{10} + 500464443061 T^{12} - 11090716582 p^{2} T^{14} + 201141458 p^{4} T^{16} - 2923858 p^{6} T^{18} + 32757 p^{8} T^{20} - 254 p^{10} T^{22} + p^{12} T^{24} \) | |
| 43 | \( 1 - 20 T + 200 T^{2} - 924 T^{3} - 1770 T^{4} + 46524 T^{5} - 149592 T^{6} - 1971628 T^{7} + 27832799 T^{8} - 162350984 T^{9} + 319707600 T^{10} + 2960692776 T^{11} - 33765637196 T^{12} + 2960692776 p T^{13} + 319707600 p^{2} T^{14} - 162350984 p^{3} T^{15} + 27832799 p^{4} T^{16} - 1971628 p^{5} T^{17} - 149592 p^{6} T^{18} + 46524 p^{7} T^{19} - 1770 p^{8} T^{20} - 924 p^{9} T^{21} + 200 p^{10} T^{22} - 20 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( ( 1 + 8 T + 147 T^{2} + 830 T^{3} + 11506 T^{4} + 50068 T^{5} + 588553 T^{6} + 50068 p T^{7} + 11506 p^{2} T^{8} + 830 p^{3} T^{9} + 147 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 53 | \( 1 + 16 T + 128 T^{2} + 1096 T^{3} + 586 T^{4} - 82200 T^{5} - 789600 T^{6} - 7806032 T^{7} - 45654497 T^{8} + 14272960 T^{9} + 1357857504 T^{10} + 20288254256 T^{11} + 219339685132 T^{12} + 20288254256 p T^{13} + 1357857504 p^{2} T^{14} + 14272960 p^{3} T^{15} - 45654497 p^{4} T^{16} - 7806032 p^{5} T^{17} - 789600 p^{6} T^{18} - 82200 p^{7} T^{19} + 586 p^{8} T^{20} + 1096 p^{9} T^{21} + 128 p^{10} T^{22} + 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 8 T + 32 T^{2} + 312 T^{3} - 8214 T^{4} + 37896 T^{5} + 8352 T^{6} - 1753848 T^{7} + 32803743 T^{8} - 128318160 T^{9} + 216281664 T^{10} + 6242043056 T^{11} - 111681131956 T^{12} + 6242043056 p T^{13} + 216281664 p^{2} T^{14} - 128318160 p^{3} T^{15} + 32803743 p^{4} T^{16} - 1753848 p^{5} T^{17} + 8352 p^{6} T^{18} + 37896 p^{7} T^{19} - 8214 p^{8} T^{20} + 312 p^{9} T^{21} + 32 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 12 T + 72 T^{2} - 436 T^{3} + 10182 T^{4} - 96692 T^{5} + 522248 T^{6} - 962348 T^{7} - 2556417 T^{8} - 61735480 T^{9} + 701613808 T^{10} + 10527482232 T^{11} - 198377340076 T^{12} + 10527482232 p T^{13} + 701613808 p^{2} T^{14} - 61735480 p^{3} T^{15} - 2556417 p^{4} T^{16} - 962348 p^{5} T^{17} + 522248 p^{6} T^{18} - 96692 p^{7} T^{19} + 10182 p^{8} T^{20} - 436 p^{9} T^{21} + 72 p^{10} T^{22} - 12 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 4 T + 8 T^{2} + 76 T^{3} - 1562 T^{4} + 12052 T^{5} + 63592 T^{6} + 2247636 T^{7} - 19830089 T^{8} - 229054608 T^{9} - 414801328 T^{10} - 5101901080 T^{11} + 60684024232 T^{12} - 5101901080 p T^{13} - 414801328 p^{2} T^{14} - 229054608 p^{3} T^{15} - 19830089 p^{4} T^{16} + 2247636 p^{5} T^{17} + 63592 p^{6} T^{18} + 12052 p^{7} T^{19} - 1562 p^{8} T^{20} + 76 p^{9} T^{21} + 8 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 - 586 T^{2} + 166561 T^{4} - 30467270 T^{6} + 4018404002 T^{8} - 405369799706 T^{10} + 32251408969501 T^{12} - 405369799706 p^{2} T^{14} + 4018404002 p^{4} T^{16} - 30467270 p^{6} T^{18} + 166561 p^{8} T^{20} - 586 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( 1 - 422 T^{2} + 93189 T^{4} - 13924530 T^{6} + 1577319194 T^{8} - 145394149854 T^{10} + 11405608578077 T^{12} - 145394149854 p^{2} T^{14} + 1577319194 p^{4} T^{16} - 13924530 p^{6} T^{18} + 93189 p^{8} T^{20} - 422 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( ( 1 + 2 T + 368 T^{2} + 1126 T^{3} + 60155 T^{4} + 204124 T^{5} + 5897072 T^{6} + 204124 p T^{7} + 60155 p^{2} T^{8} + 1126 p^{3} T^{9} + 368 p^{4} T^{10} + 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 - 28 T + 392 T^{2} - 5876 T^{3} + 84726 T^{4} - 894300 T^{5} + 9091496 T^{6} - 99906012 T^{7} + 883151735 T^{8} - 7070500512 T^{9} + 68426415632 T^{10} - 574097189672 T^{11} + 4396159810696 T^{12} - 574097189672 p T^{13} + 68426415632 p^{2} T^{14} - 7070500512 p^{3} T^{15} + 883151735 p^{4} T^{16} - 99906012 p^{5} T^{17} + 9091496 p^{6} T^{18} - 894300 p^{7} T^{19} + 84726 p^{8} T^{20} - 5876 p^{9} T^{21} + 392 p^{10} T^{22} - 28 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 636 T^{2} + 200302 T^{4} - 41757740 T^{6} + 6485840655 T^{8} - 793818201688 T^{10} + 78404791964292 T^{12} - 793818201688 p^{2} T^{14} + 6485840655 p^{4} T^{16} - 41757740 p^{6} T^{18} + 200302 p^{8} T^{20} - 636 p^{10} T^{22} + p^{12} T^{24} \) | |
| 97 | \( ( 1 - 18 T + 446 T^{2} - 6848 T^{3} + 95053 T^{4} - 1194926 T^{5} + 11749682 T^{6} - 1194926 p T^{7} + 95053 p^{2} T^{8} - 6848 p^{3} T^{9} + 446 p^{4} T^{10} - 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
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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
−3.87557437151970181617503925616, −3.83606431292845019629782321294, −3.81736409660597829717095700935, −3.71695112095465242551714340924, −3.45007524292191615862839947186, −3.41355724765996220278334269970, −3.17806668114539975544731516127, −3.09173358998269879976215077889, −3.02976387776582137422094243857, −2.98487059449252050967082662796, −2.74559376709046905725743728081, −2.63362742949133578597468971440, −2.40091883385074974794375894198, −2.39511395399818928211086739556, −2.33977441100508406934107123901, −2.14647611322891899249383761149, −2.09662749487150721203025885434, −1.89793299546158056440045377543, −1.89314453486748937723889002260, −1.24237843368814889930926872295, −1.12437948643308533121759092715, −0.983335368265266889618073751129, −0.981301979300453239216381280897, −0.76512604202615608171771781712, −0.51860020902536311059119138090, 0.51860020902536311059119138090, 0.76512604202615608171771781712, 0.981301979300453239216381280897, 0.983335368265266889618073751129, 1.12437948643308533121759092715, 1.24237843368814889930926872295, 1.89314453486748937723889002260, 1.89793299546158056440045377543, 2.09662749487150721203025885434, 2.14647611322891899249383761149, 2.33977441100508406934107123901, 2.39511395399818928211086739556, 2.40091883385074974794375894198, 2.63362742949133578597468971440, 2.74559376709046905725743728081, 2.98487059449252050967082662796, 3.02976387776582137422094243857, 3.09173358998269879976215077889, 3.17806668114539975544731516127, 3.41355724765996220278334269970, 3.45007524292191615862839947186, 3.71695112095465242551714340924, 3.81736409660597829717095700935, 3.83606431292845019629782321294, 3.87557437151970181617503925616
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.