Dirichlet series
| L(s) = 1 | + 3-s + 3·5-s + 8·7-s + 10·9-s + 4·11-s − 7·13-s + 3·15-s + 7·17-s − 3·19-s + 8·21-s − 36·23-s + 3·25-s + 6·27-s + 13·29-s − 2·31-s + 4·33-s + 24·35-s − 22·37-s − 7·39-s + 7·41-s − 6·43-s + 30·45-s + 2·47-s + 33·49-s + 7·51-s + 3·53-s + 12·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 3.02·7-s + 10/3·9-s + 1.20·11-s − 1.94·13-s + 0.774·15-s + 1.69·17-s − 0.688·19-s + 1.74·21-s − 7.50·23-s + 3/5·25-s + 1.15·27-s + 2.41·29-s − 0.359·31-s + 0.696·33-s + 4.05·35-s − 3.61·37-s − 1.12·39-s + 1.09·41-s − 0.914·43-s + 4.47·45-s + 0.291·47-s + 33/7·49-s + 0.980·51-s + 0.412·53-s + 1.61·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{12} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 5^{12} \cdot 11^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{48} \cdot 5^{12} \cdot 11^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.44916\times 10^{10}\) |
| Root analytic conductor: | \(2.65081\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{48} \cdot 5^{12} \cdot 11^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(9.849419419\) |
| \(L(\frac12)\) | \(\approx\) | \(9.849419419\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{3} \) | |
| 11 | \( 1 - 4 T + 12 T^{2} - 53 T^{3} + 208 T^{4} - 978 T^{5} + 3423 T^{6} - 978 p T^{7} + 208 p^{2} T^{8} - 53 p^{3} T^{9} + 12 p^{4} T^{10} - 4 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 3 | \( 1 - T - p^{2} T^{2} + 13 T^{3} + 43 T^{4} - 52 T^{5} - 199 T^{6} + 124 T^{7} + 964 T^{8} - 358 T^{9} - 391 p^{2} T^{10} + 70 p^{2} T^{11} + 10735 T^{12} + 70 p^{3} T^{13} - 391 p^{4} T^{14} - 358 p^{3} T^{15} + 964 p^{4} T^{16} + 124 p^{5} T^{17} - 199 p^{6} T^{18} - 52 p^{7} T^{19} + 43 p^{8} T^{20} + 13 p^{9} T^{21} - p^{12} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( 1 - 8 T + 31 T^{2} - 122 T^{3} + 536 T^{4} - 2106 T^{5} + 7188 T^{6} - 22510 T^{7} + 10117 p T^{8} - 221121 T^{9} + 653090 T^{10} - 1767427 T^{11} + 4597406 T^{12} - 1767427 p T^{13} + 653090 p^{2} T^{14} - 221121 p^{3} T^{15} + 10117 p^{5} T^{16} - 22510 p^{5} T^{17} + 7188 p^{6} T^{18} - 2106 p^{7} T^{19} + 536 p^{8} T^{20} - 122 p^{9} T^{21} + 31 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 7 T + 32 T^{2} + 119 T^{3} + 541 T^{4} + 1481 T^{5} + 3243 T^{6} + 1961 T^{7} - 13008 T^{8} - 203587 T^{9} - 775144 T^{10} - 3355845 T^{11} - 10020050 T^{12} - 3355845 p T^{13} - 775144 p^{2} T^{14} - 203587 p^{3} T^{15} - 13008 p^{4} T^{16} + 1961 p^{5} T^{17} + 3243 p^{6} T^{18} + 1481 p^{7} T^{19} + 541 p^{8} T^{20} + 119 p^{9} T^{21} + 32 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 7 T + p T^{2} - 104 T^{3} + 633 T^{4} + 1476 T^{5} - 11797 T^{6} + 206 T^{7} - 144458 T^{8} + 831746 T^{9} + 5257183 T^{10} - 24093207 T^{11} + 28389639 T^{12} - 24093207 p T^{13} + 5257183 p^{2} T^{14} + 831746 p^{3} T^{15} - 144458 p^{4} T^{16} + 206 p^{5} T^{17} - 11797 p^{6} T^{18} + 1476 p^{7} T^{19} + 633 p^{8} T^{20} - 104 p^{9} T^{21} + p^{11} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 3 T - 6 T^{2} - p T^{3} + 435 T^{4} + 2591 T^{5} - 2183 T^{6} - 40794 T^{7} - 65696 T^{8} + 15899 p T^{9} + 22255 T^{10} - 14121802 T^{11} - 97838007 T^{12} - 14121802 p T^{13} + 22255 p^{2} T^{14} + 15899 p^{4} T^{15} - 65696 p^{4} T^{16} - 40794 p^{5} T^{17} - 2183 p^{6} T^{18} + 2591 p^{7} T^{19} + 435 p^{8} T^{20} - p^{10} T^{21} - 6 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( ( 1 + 18 T + 188 T^{2} + 1501 T^{3} + 10104 T^{4} + 58774 T^{5} + 299719 T^{6} + 58774 p T^{7} + 10104 p^{2} T^{8} + 1501 p^{3} T^{9} + 188 p^{4} T^{10} + 18 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 29 | \( 1 - 13 T - 18 T^{2} + 933 T^{3} - 1455 T^{4} - 45511 T^{5} + 188151 T^{6} + 1535916 T^{7} - 10865830 T^{8} - 35417083 T^{9} + 435952699 T^{10} + 429954742 T^{11} - 14340814195 T^{12} + 429954742 p T^{13} + 435952699 p^{2} T^{14} - 35417083 p^{3} T^{15} - 10865830 p^{4} T^{16} + 1535916 p^{5} T^{17} + 188151 p^{6} T^{18} - 45511 p^{7} T^{19} - 1455 p^{8} T^{20} + 933 p^{9} T^{21} - 18 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 2 T - 32 T^{2} - 147 T^{3} + 585 T^{4} - 10798 T^{5} - 12064 T^{6} + 442166 T^{7} + 2054167 T^{8} - 5639388 T^{9} + 47633255 T^{10} - 193657875 T^{11} - 2835393253 T^{12} - 193657875 p T^{13} + 47633255 p^{2} T^{14} - 5639388 p^{3} T^{15} + 2054167 p^{4} T^{16} + 442166 p^{5} T^{17} - 12064 p^{6} T^{18} - 10798 p^{7} T^{19} + 585 p^{8} T^{20} - 147 p^{9} T^{21} - 32 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 22 T + 180 T^{2} + 698 T^{3} + 2410 T^{4} + 14428 T^{5} + 70467 T^{6} + 429482 T^{7} + 525048 T^{8} - 35878288 T^{9} - 369383214 T^{10} - 2023911902 T^{11} - 10623107595 T^{12} - 2023911902 p T^{13} - 369383214 p^{2} T^{14} - 35878288 p^{3} T^{15} + 525048 p^{4} T^{16} + 429482 p^{5} T^{17} + 70467 p^{6} T^{18} + 14428 p^{7} T^{19} + 2410 p^{8} T^{20} + 698 p^{9} T^{21} + 180 p^{10} T^{22} + 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 7 T + 6 T^{2} - 182 T^{3} - 255 T^{4} + 13336 T^{5} + 37380 T^{6} - 196917 T^{7} + 940756 T^{8} - 36594487 T^{9} - 164631 T^{10} + 763640797 T^{11} + 2385234022 T^{12} + 763640797 p T^{13} - 164631 p^{2} T^{14} - 36594487 p^{3} T^{15} + 940756 p^{4} T^{16} - 196917 p^{5} T^{17} + 37380 p^{6} T^{18} + 13336 p^{7} T^{19} - 255 p^{8} T^{20} - 182 p^{9} T^{21} + 6 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( ( 1 + 3 T + 95 T^{2} - 260 T^{3} + 4732 T^{4} - 17257 T^{5} + 289940 T^{6} - 17257 p T^{7} + 4732 p^{2} T^{8} - 260 p^{3} T^{9} + 95 p^{4} T^{10} + 3 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 - 2 T - 100 T^{2} + 465 T^{3} + 3883 T^{4} - 56166 T^{5} + 52458 T^{6} + 3148400 T^{7} - 13278623 T^{8} - 89904814 T^{9} + 1018574969 T^{10} + 991423921 T^{11} - 53423685691 T^{12} + 991423921 p T^{13} + 1018574969 p^{2} T^{14} - 89904814 p^{3} T^{15} - 13278623 p^{4} T^{16} + 3148400 p^{5} T^{17} + 52458 p^{6} T^{18} - 56166 p^{7} T^{19} + 3883 p^{8} T^{20} + 465 p^{9} T^{21} - 100 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 3 T - 104 T^{2} - 322 T^{3} + 12013 T^{4} - 1098 T^{5} - 700689 T^{6} - 973236 T^{7} + 67611734 T^{8} - 38761152 T^{9} - 3362852859 T^{10} - 4072754491 T^{11} + 226784685765 T^{12} - 4072754491 p T^{13} - 3362852859 p^{2} T^{14} - 38761152 p^{3} T^{15} + 67611734 p^{4} T^{16} - 973236 p^{5} T^{17} - 700689 p^{6} T^{18} - 1098 p^{7} T^{19} + 12013 p^{8} T^{20} - 322 p^{9} T^{21} - 104 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 19 T + 114 T^{2} - 844 T^{3} - 18205 T^{4} - 94472 T^{5} + 388105 T^{6} + 8056646 T^{7} + 35220186 T^{8} - 131818866 T^{9} - 1974980209 T^{10} - 4343767901 T^{11} + 29551371817 T^{12} - 4343767901 p T^{13} - 1974980209 p^{2} T^{14} - 131818866 p^{3} T^{15} + 35220186 p^{4} T^{16} + 8056646 p^{5} T^{17} + 388105 p^{6} T^{18} - 94472 p^{7} T^{19} - 18205 p^{8} T^{20} - 844 p^{9} T^{21} + 114 p^{10} T^{22} + 19 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 22 T + 199 T^{2} - 128 T^{3} - 22294 T^{4} + 326058 T^{5} - 1845298 T^{6} - 7876470 T^{7} + 242830107 T^{8} - 2052940678 T^{9} + 4360347054 T^{10} + 98430831970 T^{11} - 1263891214637 T^{12} + 98430831970 p T^{13} + 4360347054 p^{2} T^{14} - 2052940678 p^{3} T^{15} + 242830107 p^{4} T^{16} - 7876470 p^{5} T^{17} - 1845298 p^{6} T^{18} + 326058 p^{7} T^{19} - 22294 p^{8} T^{20} - 128 p^{9} T^{21} + 199 p^{10} T^{22} - 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( ( 1 + 11 T + 336 T^{2} + 2995 T^{3} + 51392 T^{4} + 365850 T^{5} + 4451546 T^{6} + 365850 p T^{7} + 51392 p^{2} T^{8} + 2995 p^{3} T^{9} + 336 p^{4} T^{10} + 11 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( 1 + 44 T + 930 T^{2} + 14416 T^{3} + 206018 T^{4} + 2654312 T^{5} + 29171195 T^{6} + 296751638 T^{7} + 2979783678 T^{8} + 27771843578 T^{9} + 240414887090 T^{10} + 2116788970022 T^{11} + 18489055172657 T^{12} + 2116788970022 p T^{13} + 240414887090 p^{2} T^{14} + 27771843578 p^{3} T^{15} + 2979783678 p^{4} T^{16} + 296751638 p^{5} T^{17} + 29171195 p^{6} T^{18} + 2654312 p^{7} T^{19} + 206018 p^{8} T^{20} + 14416 p^{9} T^{21} + 930 p^{10} T^{22} + 44 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 17 T + 37 T^{2} + 936 T^{3} + 1131 T^{4} - 34086 T^{5} - 1348517 T^{6} + 15164264 T^{7} + 21210462 T^{8} - 381735628 T^{9} - 2672916579 T^{10} - 32433327665 T^{11} + 908514004435 T^{12} - 32433327665 p T^{13} - 2672916579 p^{2} T^{14} - 381735628 p^{3} T^{15} + 21210462 p^{4} T^{16} + 15164264 p^{5} T^{17} - 1348517 p^{6} T^{18} - 34086 p^{7} T^{19} + 1131 p^{8} T^{20} + 936 p^{9} T^{21} + 37 p^{10} T^{22} - 17 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 43 T + 737 T^{2} - 5687 T^{3} + 9175 T^{4} + 15604 T^{5} + 3910371 T^{6} - 71436854 T^{7} + 574044690 T^{8} - 2751271868 T^{9} + 19546079309 T^{10} - 298266633288 T^{11} + 3398208440155 T^{12} - 298266633288 p T^{13} + 19546079309 p^{2} T^{14} - 2751271868 p^{3} T^{15} + 574044690 p^{4} T^{16} - 71436854 p^{5} T^{17} + 3910371 p^{6} T^{18} + 15604 p^{7} T^{19} + 9175 p^{8} T^{20} - 5687 p^{9} T^{21} + 737 p^{10} T^{22} - 43 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 15 T + 294 T^{2} - 3582 T^{3} + 49733 T^{4} - 6852 p T^{5} + 6995797 T^{6} - 79167644 T^{7} + 872334228 T^{8} - 9224897458 T^{9} + 88237369339 T^{10} - 861626080359 T^{11} + 7693956809419 T^{12} - 861626080359 p T^{13} + 88237369339 p^{2} T^{14} - 9224897458 p^{3} T^{15} + 872334228 p^{4} T^{16} - 79167644 p^{5} T^{17} + 6995797 p^{6} T^{18} - 6852 p^{8} T^{19} + 49733 p^{8} T^{20} - 3582 p^{9} T^{21} + 294 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( ( 1 - T + 115 T^{2} - 899 T^{3} + 7226 T^{4} - 16021 T^{5} + 1054107 T^{6} - 16021 p T^{7} + 7226 p^{2} T^{8} - 899 p^{3} T^{9} + 115 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 97 | \( 1 + 20 T - 226 T^{2} - 7588 T^{3} + 714 T^{4} + 1270696 T^{5} + 5314879 T^{6} - 131460052 T^{7} - 984200832 T^{8} + 9951735072 T^{9} + 128753922634 T^{10} - 384950631020 T^{11} - 14003226843683 T^{12} - 384950631020 p T^{13} + 128753922634 p^{2} T^{14} + 9951735072 p^{3} T^{15} - 984200832 p^{4} T^{16} - 131460052 p^{5} T^{17} + 5314879 p^{6} T^{18} + 1270696 p^{7} T^{19} + 714 p^{8} T^{20} - 7588 p^{9} T^{21} - 226 p^{10} T^{22} + 20 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.20334696079967761760642491912, −3.16802641099933992243000004420, −3.03984619614296848912605785134, −2.98878247903217330553046448882, −2.98794694508182644931557561963, −2.64505738687594244955938383382, −2.57234131966681635926267957078, −2.45136883825335386623294010873, −2.25296164350541388779498618465, −2.12477193706392703618471658325, −2.09450188815311782842243725393, −2.05814959903646051779388004882, −2.02611006162754131912847644130, −1.86512736383743533312782694323, −1.76755120625221051070723366666, −1.68403645475564312576439353745, −1.66486643679667480871358534222, −1.57458018079066076803247792964, −1.46636079090880496786433213120, −1.22484325294029935441966972292, −0.996529407774496632244355020546, −0.856620744019766690853499103171, −0.847435964789743550745600213937, −0.39196721906961579419157597660, −0.15466919875236229759130851889, 0.15466919875236229759130851889, 0.39196721906961579419157597660, 0.847435964789743550745600213937, 0.856620744019766690853499103171, 0.996529407774496632244355020546, 1.22484325294029935441966972292, 1.46636079090880496786433213120, 1.57458018079066076803247792964, 1.66486643679667480871358534222, 1.68403645475564312576439353745, 1.76755120625221051070723366666, 1.86512736383743533312782694323, 2.02611006162754131912847644130, 2.05814959903646051779388004882, 2.09450188815311782842243725393, 2.12477193706392703618471658325, 2.25296164350541388779498618465, 2.45136883825335386623294010873, 2.57234131966681635926267957078, 2.64505738687594244955938383382, 2.98794694508182644931557561963, 2.98878247903217330553046448882, 3.03984619614296848912605785134, 3.16802641099933992243000004420, 3.20334696079967761760642491912
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.