Dirichlet series
| L(s) = 1 | + 5·4-s − 8·13-s + 8·16-s + 12·17-s + 4·23-s + 22·25-s − 8·29-s + 12·43-s + 58·49-s − 40·52-s + 20·53-s − 12·61-s + 60·68-s − 48·79-s + 20·92-s + 110·100-s + 8·101-s + 32·103-s − 16·107-s − 16·113-s − 40·116-s − 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + ⋯ |
| L(s) = 1 | + 5/2·4-s − 2.21·13-s + 2·16-s + 2.91·17-s + 0.834·23-s + 22/5·25-s − 1.48·29-s + 1.82·43-s + 58/7·49-s − 5.54·52-s + 2.74·53-s − 1.53·61-s + 7.27·68-s − 5.40·79-s + 2.08·92-s + 11·100-s + 0.796·101-s + 3.15·103-s − 1.54·107-s − 1.50·113-s − 3.71·116-s − 0.545·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 11^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 11^{12} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{24} \cdot 11^{12} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.38761\times 10^{12}\) |
| Root analytic conductor: | \(3.20573\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{24} \cdot 11^{12} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(54.52387000\) |
| \(L(\frac12)\) | \(\approx\) | \(54.52387000\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 11 | \( ( 1 + T^{2} )^{6} \) | |
| 13 | \( 1 + 8 T + 20 T^{2} + 40 T^{3} + 287 T^{4} + 1328 T^{5} + 4584 T^{6} + 1328 p T^{7} + 287 p^{2} T^{8} + 40 p^{3} T^{9} + 20 p^{4} T^{10} + 8 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 2 | \( 1 - 5 T^{2} + 17 T^{4} - 45 T^{6} + 101 T^{8} - 51 p^{2} T^{10} + 103 p^{2} T^{12} - 51 p^{4} T^{14} + 101 p^{4} T^{16} - 45 p^{6} T^{18} + 17 p^{8} T^{20} - 5 p^{10} T^{22} + p^{12} T^{24} \) |
| 5 | \( 1 - 22 T^{2} + 287 T^{4} - 2722 T^{6} + 4151 p T^{8} - 132344 T^{10} + 715994 T^{12} - 132344 p^{2} T^{14} + 4151 p^{5} T^{16} - 2722 p^{6} T^{18} + 287 p^{8} T^{20} - 22 p^{10} T^{22} + p^{12} T^{24} \) | |
| 7 | \( 1 - 58 T^{2} + 1661 T^{4} - 30904 T^{6} + 414824 T^{8} - 4224114 T^{10} + 33423324 T^{12} - 4224114 p^{2} T^{14} + 414824 p^{4} T^{16} - 30904 p^{6} T^{18} + 1661 p^{8} T^{20} - 58 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( ( 1 - 6 T + 4 p T^{2} - 234 T^{3} + 1455 T^{4} - 2832 T^{5} + 19896 T^{6} - 2832 p T^{7} + 1455 p^{2} T^{8} - 234 p^{3} T^{9} + 4 p^{5} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 19 | \( 1 - 110 T^{2} + 6849 T^{4} - 295280 T^{6} + 9707444 T^{8} - 252675522 T^{10} + 5325171464 T^{12} - 252675522 p^{2} T^{14} + 9707444 p^{4} T^{16} - 295280 p^{6} T^{18} + 6849 p^{8} T^{20} - 110 p^{10} T^{22} + p^{12} T^{24} \) | |
| 23 | \( ( 1 - 2 T + 40 T^{2} + 144 T^{3} + 949 T^{4} + 2476 T^{5} + 43293 T^{6} + 2476 p T^{7} + 949 p^{2} T^{8} + 144 p^{3} T^{9} + 40 p^{4} T^{10} - 2 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 29 | \( ( 1 + 4 T + 132 T^{2} + 532 T^{3} + 8111 T^{4} + 29224 T^{5} + 297096 T^{6} + 29224 p T^{7} + 8111 p^{2} T^{8} + 532 p^{3} T^{9} + 132 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 31 | \( 1 - 210 T^{2} + 21511 T^{4} - 1461586 T^{6} + 75243131 T^{8} - 3121766748 T^{10} + 106535910906 T^{12} - 3121766748 p^{2} T^{14} + 75243131 p^{4} T^{16} - 1461586 p^{6} T^{18} + 21511 p^{8} T^{20} - 210 p^{10} T^{22} + p^{12} T^{24} \) | |
| 37 | \( 1 - 218 T^{2} + 26207 T^{4} - 2191994 T^{6} + 139565323 T^{8} - 7059997468 T^{10} + 289296274538 T^{12} - 7059997468 p^{2} T^{14} + 139565323 p^{4} T^{16} - 2191994 p^{6} T^{18} + 26207 p^{8} T^{20} - 218 p^{10} T^{22} + p^{12} T^{24} \) | |
| 41 | \( 1 - 350 T^{2} + 60137 T^{4} - 6677264 T^{6} + 532053524 T^{8} - 31989562698 T^{10} + 1486526234040 T^{12} - 31989562698 p^{2} T^{14} + 532053524 p^{4} T^{16} - 6677264 p^{6} T^{18} + 60137 p^{8} T^{20} - 350 p^{10} T^{22} + p^{12} T^{24} \) | |
| 43 | \( ( 1 - 6 T + 4 p T^{2} - 1150 T^{3} + 14063 T^{4} - 92496 T^{5} + 732408 T^{6} - 92496 p T^{7} + 14063 p^{2} T^{8} - 1150 p^{3} T^{9} + 4 p^{5} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 - 344 T^{2} + 56746 T^{4} - 6111480 T^{6} + 490453519 T^{8} - 31258355888 T^{10} + 1622912397580 T^{12} - 31258355888 p^{2} T^{14} + 490453519 p^{4} T^{16} - 6111480 p^{6} T^{18} + 56746 p^{8} T^{20} - 344 p^{10} T^{22} + p^{12} T^{24} \) | |
| 53 | \( ( 1 - 10 T + 217 T^{2} - 1476 T^{3} + 19432 T^{4} - 99010 T^{5} + 1144464 T^{6} - 99010 p T^{7} + 19432 p^{2} T^{8} - 1476 p^{3} T^{9} + 217 p^{4} T^{10} - 10 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 - 274 T^{2} + 45847 T^{4} - 5510178 T^{6} + 520879275 T^{8} - 40353433084 T^{10} + 2597499085498 T^{12} - 40353433084 p^{2} T^{14} + 520879275 p^{4} T^{16} - 5510178 p^{6} T^{18} + 45847 p^{8} T^{20} - 274 p^{10} T^{22} + p^{12} T^{24} \) | |
| 61 | \( ( 1 + 6 T + 136 T^{2} + 1330 T^{3} + 13463 T^{4} + 91968 T^{5} + 1111632 T^{6} + 91968 p T^{7} + 13463 p^{2} T^{8} + 1330 p^{3} T^{9} + 136 p^{4} T^{10} + 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( 1 - 294 T^{2} + 42823 T^{4} - 4204714 T^{6} + 316149251 T^{8} - 20061645312 T^{10} + 1266508121130 T^{12} - 20061645312 p^{2} T^{14} + 316149251 p^{4} T^{16} - 4204714 p^{6} T^{18} + 42823 p^{8} T^{20} - 294 p^{10} T^{22} + p^{12} T^{24} \) | |
| 71 | \( 1 - 354 T^{2} + 74695 T^{4} - 11079698 T^{6} + 1290715451 T^{8} - 121403301996 T^{10} + 9475711215738 T^{12} - 121403301996 p^{2} T^{14} + 1290715451 p^{4} T^{16} - 11079698 p^{6} T^{18} + 74695 p^{8} T^{20} - 354 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( 1 - 642 T^{2} + 195557 T^{4} - 37725800 T^{6} + 5188402488 T^{8} - 541684076186 T^{10} + 44392042334828 T^{12} - 541684076186 p^{2} T^{14} + 5188402488 p^{4} T^{16} - 37725800 p^{6} T^{18} + 195557 p^{8} T^{20} - 642 p^{10} T^{22} + p^{12} T^{24} \) | |
| 79 | \( ( 1 + 24 T + 604 T^{2} + 9384 T^{3} + 133959 T^{4} + 1474528 T^{5} + 14612440 T^{6} + 1474528 p T^{7} + 133959 p^{2} T^{8} + 9384 p^{3} T^{9} + 604 p^{4} T^{10} + 24 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 83 | \( 1 - 518 T^{2} + 131393 T^{4} - 21504800 T^{6} + 2572651652 T^{8} - 248777467218 T^{10} + 21398981031480 T^{12} - 248777467218 p^{2} T^{14} + 2572651652 p^{4} T^{16} - 21504800 p^{6} T^{18} + 131393 p^{8} T^{20} - 518 p^{10} T^{22} + p^{12} T^{24} \) | |
| 89 | \( 1 - 502 T^{2} + 141807 T^{4} - 314402 p T^{6} + 4226117411 T^{8} - 509690782104 T^{10} + 50083832568410 T^{12} - 509690782104 p^{2} T^{14} + 4226117411 p^{4} T^{16} - 314402 p^{7} T^{18} + 141807 p^{8} T^{20} - 502 p^{10} T^{22} + p^{12} T^{24} \) | |
| 97 | \( 1 - 810 T^{2} + 313519 T^{4} - 77379274 T^{6} + 13723791611 T^{8} - 1866852082188 T^{10} + 201893417771178 T^{12} - 1866852082188 p^{2} T^{14} + 13723791611 p^{4} T^{16} - 77379274 p^{6} T^{18} + 313519 p^{8} T^{20} - 810 p^{10} T^{22} + p^{12} T^{24} \) | |
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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
−3.07327396791489707938686975426, −2.99408784193540078084965898108, −2.84215560511707990008706510133, −2.71540874845043620743783380802, −2.55488223619835968058291721731, −2.50918909083484773840105006472, −2.48727865008275262152218316437, −2.44113192369950114020042883266, −2.34801209832211700363604951660, −2.19576921972102966321698342486, −2.16764797794204544474564659781, −2.02314984259529910038793183287, −1.90176322205685709169947480218, −1.83061261231313666541068925717, −1.81797221231566999110670515177, −1.37918833134754924704769678630, −1.26015678888744697163518272561, −1.25581255525817886923882214843, −1.10200889168440077542488724435, −1.07497177439335292986185827634, −0.948476672721698584726236904451, −0.793543984062227174081566668481, −0.63458795069577072491283658230, −0.38380028135966411125506556498, −0.33158020533583318759748743978, 0.33158020533583318759748743978, 0.38380028135966411125506556498, 0.63458795069577072491283658230, 0.793543984062227174081566668481, 0.948476672721698584726236904451, 1.07497177439335292986185827634, 1.10200889168440077542488724435, 1.25581255525817886923882214843, 1.26015678888744697163518272561, 1.37918833134754924704769678630, 1.81797221231566999110670515177, 1.83061261231313666541068925717, 1.90176322205685709169947480218, 2.02314984259529910038793183287, 2.16764797794204544474564659781, 2.19576921972102966321698342486, 2.34801209832211700363604951660, 2.44113192369950114020042883266, 2.48727865008275262152218316437, 2.50918909083484773840105006472, 2.55488223619835968058291721731, 2.71540874845043620743783380802, 2.84215560511707990008706510133, 2.99408784193540078084965898108, 3.07327396791489707938686975426
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.