Dirichlet series
| L(s) = 1 | + 7·2-s + 19·4-s + 7·5-s − 15·7-s + 20·8-s + 49·10-s − 12·11-s − 11·13-s − 105·14-s − 12·16-s + 33·17-s − 24·19-s + 133·20-s − 84·22-s + 9·23-s − 77·26-s − 285·28-s + 16·29-s + 31-s − 57·32-s + 231·34-s − 105·35-s − 6·37-s − 168·38-s + 140·40-s + 21·41-s − 39·43-s + ⋯ |
| L(s) = 1 | + 4.94·2-s + 19/2·4-s + 3.13·5-s − 5.66·7-s + 7.07·8-s + 15.4·10-s − 3.61·11-s − 3.05·13-s − 28.0·14-s − 3·16-s + 8.00·17-s − 5.50·19-s + 29.7·20-s − 17.9·22-s + 1.87·23-s − 15.1·26-s − 53.8·28-s + 2.97·29-s + 0.179·31-s − 10.0·32-s + 39.6·34-s − 17.7·35-s − 0.986·37-s − 27.2·38-s + 22.1·40-s + 3.27·41-s − 5.94·43-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 11^{12} \cdot 61^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{24} \cdot 11^{12} \cdot 61^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(3^{24} \cdot 11^{12} \cdot 61^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.58090\times 10^{20}\) |
| Root analytic conductor: | \(6.94418\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 3^{24} \cdot 11^{12} \cdot 61^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(79.88166540\) |
| \(L(\frac12)\) | \(\approx\) | \(79.88166540\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 + T )^{12} \) | |
| 61 | \( ( 1 + T )^{12} \) | |
| good | 2 | \( 1 - 7 T + 15 p T^{2} - 97 T^{3} + 261 T^{4} - 611 T^{5} + 643 p T^{6} - 1241 p T^{7} + 4459 T^{8} - 7521 T^{9} + 12001 T^{10} - 18203 T^{11} + 26347 T^{12} - 18203 p T^{13} + 12001 p^{2} T^{14} - 7521 p^{3} T^{15} + 4459 p^{4} T^{16} - 1241 p^{6} T^{17} + 643 p^{7} T^{18} - 611 p^{7} T^{19} + 261 p^{8} T^{20} - 97 p^{9} T^{21} + 15 p^{11} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) |
| 5 | \( 1 - 7 T + 49 T^{2} - 9 p^{2} T^{3} + 972 T^{4} - 696 p T^{5} + 11659 T^{6} - 35101 T^{7} + 19886 p T^{8} - 262799 T^{9} + 659489 T^{10} - 1575031 T^{11} + 3598106 T^{12} - 1575031 p T^{13} + 659489 p^{2} T^{14} - 262799 p^{3} T^{15} + 19886 p^{5} T^{16} - 35101 p^{5} T^{17} + 11659 p^{6} T^{18} - 696 p^{8} T^{19} + 972 p^{8} T^{20} - 9 p^{11} T^{21} + 49 p^{10} T^{22} - 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 7 | \( 1 + 15 T + 139 T^{2} + 948 T^{3} + 5305 T^{4} + 25518 T^{5} + 109397 T^{6} + 426094 T^{7} + 1529603 T^{8} + 728057 p T^{9} + 15850126 T^{10} + 46134877 T^{11} + 126000578 T^{12} + 46134877 p T^{13} + 15850126 p^{2} T^{14} + 728057 p^{4} T^{15} + 1529603 p^{4} T^{16} + 426094 p^{5} T^{17} + 109397 p^{6} T^{18} + 25518 p^{7} T^{19} + 5305 p^{8} T^{20} + 948 p^{9} T^{21} + 139 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 11 T + 97 T^{2} + 662 T^{3} + 4145 T^{4} + 22669 T^{5} + 115669 T^{6} + 537401 T^{7} + 2387791 T^{8} + 9912352 T^{9} + 39737149 T^{10} + 150753874 T^{11} + 556680330 T^{12} + 150753874 p T^{13} + 39737149 p^{2} T^{14} + 9912352 p^{3} T^{15} + 2387791 p^{4} T^{16} + 537401 p^{5} T^{17} + 115669 p^{6} T^{18} + 22669 p^{7} T^{19} + 4145 p^{8} T^{20} + 662 p^{9} T^{21} + 97 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 33 T + 597 T^{2} - 7431 T^{3} + 70404 T^{4} - 532853 T^{5} + 3320531 T^{6} - 17322548 T^{7} + 76479658 T^{8} - 288030690 T^{9} + 946301382 T^{10} - 2935275070 T^{11} + 10476783848 T^{12} - 2935275070 p T^{13} + 946301382 p^{2} T^{14} - 288030690 p^{3} T^{15} + 76479658 p^{4} T^{16} - 17322548 p^{5} T^{17} + 3320531 p^{6} T^{18} - 532853 p^{7} T^{19} + 70404 p^{8} T^{20} - 7431 p^{9} T^{21} + 597 p^{10} T^{22} - 33 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 24 T + 371 T^{2} + 4280 T^{3} + 41156 T^{4} + 341152 T^{5} + 2518088 T^{6} + 16766116 T^{7} + 102173034 T^{8} + 573298456 T^{9} + 2983989899 T^{10} + 14439165481 T^{11} + 65182491500 T^{12} + 14439165481 p T^{13} + 2983989899 p^{2} T^{14} + 573298456 p^{3} T^{15} + 102173034 p^{4} T^{16} + 16766116 p^{5} T^{17} + 2518088 p^{6} T^{18} + 341152 p^{7} T^{19} + 41156 p^{8} T^{20} + 4280 p^{9} T^{21} + 371 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 9 T + 122 T^{2} - 871 T^{3} + 8659 T^{4} - 52566 T^{5} + 409672 T^{6} - 2213499 T^{7} + 15173441 T^{8} - 73639616 T^{9} + 449170145 T^{10} - 2008306342 T^{11} + 11293340992 T^{12} - 2008306342 p T^{13} + 449170145 p^{2} T^{14} - 73639616 p^{3} T^{15} + 15173441 p^{4} T^{16} - 2213499 p^{5} T^{17} + 409672 p^{6} T^{18} - 52566 p^{7} T^{19} + 8659 p^{8} T^{20} - 871 p^{9} T^{21} + 122 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 16 T + 250 T^{2} - 2714 T^{3} + 27246 T^{4} - 229891 T^{5} + 1814447 T^{6} - 12769588 T^{7} + 85498737 T^{8} - 527376390 T^{9} + 3151032337 T^{10} - 611171962 p T^{11} + 97961804350 T^{12} - 611171962 p^{2} T^{13} + 3151032337 p^{2} T^{14} - 527376390 p^{3} T^{15} + 85498737 p^{4} T^{16} - 12769588 p^{5} T^{17} + 1814447 p^{6} T^{18} - 229891 p^{7} T^{19} + 27246 p^{8} T^{20} - 2714 p^{9} T^{21} + 250 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - T + 129 T^{2} - 543 T^{3} + 9764 T^{4} - 58349 T^{5} + 613100 T^{6} - 3780663 T^{7} + 31015429 T^{8} - 186699029 T^{9} + 40851128 p T^{10} - 7274116808 T^{11} + 42944835044 T^{12} - 7274116808 p T^{13} + 40851128 p^{3} T^{14} - 186699029 p^{3} T^{15} + 31015429 p^{4} T^{16} - 3780663 p^{5} T^{17} + 613100 p^{6} T^{18} - 58349 p^{7} T^{19} + 9764 p^{8} T^{20} - 543 p^{9} T^{21} + 129 p^{10} T^{22} - p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 6 T + 212 T^{2} + 21 p T^{3} + 20725 T^{4} + 41254 T^{5} + 1358539 T^{6} + 1119107 T^{7} + 72741900 T^{8} + 23069488 T^{9} + 3440579340 T^{10} + 894799187 T^{11} + 139567844104 T^{12} + 894799187 p T^{13} + 3440579340 p^{2} T^{14} + 23069488 p^{3} T^{15} + 72741900 p^{4} T^{16} + 1119107 p^{5} T^{17} + 1358539 p^{6} T^{18} + 41254 p^{7} T^{19} + 20725 p^{8} T^{20} + 21 p^{10} T^{21} + 212 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 21 T + 409 T^{2} - 5323 T^{3} + 65528 T^{4} - 16334 p T^{5} + 6574879 T^{6} - 57551319 T^{7} + 484655090 T^{8} - 3742091431 T^{9} + 27770380575 T^{10} - 191467662475 T^{11} + 1269193856456 T^{12} - 191467662475 p T^{13} + 27770380575 p^{2} T^{14} - 3742091431 p^{3} T^{15} + 484655090 p^{4} T^{16} - 57551319 p^{5} T^{17} + 6574879 p^{6} T^{18} - 16334 p^{8} T^{19} + 65528 p^{8} T^{20} - 5323 p^{9} T^{21} + 409 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 + 39 T + 916 T^{2} + 15723 T^{3} + 221865 T^{4} + 2696997 T^{5} + 29240744 T^{6} + 287210810 T^{7} + 2589998088 T^{8} + 21591484480 T^{9} + 167516124308 T^{10} + 1213165093292 T^{11} + 8223006197160 T^{12} + 1213165093292 p T^{13} + 167516124308 p^{2} T^{14} + 21591484480 p^{3} T^{15} + 2589998088 p^{4} T^{16} + 287210810 p^{5} T^{17} + 29240744 p^{6} T^{18} + 2696997 p^{7} T^{19} + 221865 p^{8} T^{20} + 15723 p^{9} T^{21} + 916 p^{10} T^{22} + 39 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 18 T + 379 T^{2} - 4654 T^{3} + 57946 T^{4} - 554755 T^{5} + 5371745 T^{6} - 44514567 T^{7} + 383459216 T^{8} - 2951440873 T^{9} + 502645938 p T^{10} - 168738502880 T^{11} + 1228757363190 T^{12} - 168738502880 p T^{13} + 502645938 p^{3} T^{14} - 2951440873 p^{3} T^{15} + 383459216 p^{4} T^{16} - 44514567 p^{5} T^{17} + 5371745 p^{6} T^{18} - 554755 p^{7} T^{19} + 57946 p^{8} T^{20} - 4654 p^{9} T^{21} + 379 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 14 T + 404 T^{2} - 4409 T^{3} + 75815 T^{4} - 695281 T^{5} + 9143452 T^{6} - 73386372 T^{7} + 812507082 T^{8} - 5855303311 T^{9} + 57239299361 T^{10} - 375557294264 T^{11} + 3321887781522 T^{12} - 375557294264 p T^{13} + 57239299361 p^{2} T^{14} - 5855303311 p^{3} T^{15} + 812507082 p^{4} T^{16} - 73386372 p^{5} T^{17} + 9143452 p^{6} T^{18} - 695281 p^{7} T^{19} + 75815 p^{8} T^{20} - 4409 p^{9} T^{21} + 404 p^{10} T^{22} - 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 - 23 T + 515 T^{2} - 7688 T^{3} + 104142 T^{4} - 1187224 T^{5} + 12217738 T^{6} - 114370298 T^{7} + 980246121 T^{8} - 7957083742 T^{9} + 61521602112 T^{10} - 468570698606 T^{11} + 60718073446 p T^{12} - 468570698606 p T^{13} + 61521602112 p^{2} T^{14} - 7957083742 p^{3} T^{15} + 980246121 p^{4} T^{16} - 114370298 p^{5} T^{17} + 12217738 p^{6} T^{18} - 1187224 p^{7} T^{19} + 104142 p^{8} T^{20} - 7688 p^{9} T^{21} + 515 p^{10} T^{22} - 23 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 + 449 T^{2} + 173 T^{3} + 90315 T^{4} + 119182 T^{5} + 10802408 T^{6} + 33210707 T^{7} + 873198976 T^{8} + 5211503508 T^{9} + 54633204281 T^{10} + 521654265811 T^{11} + 3370799768398 T^{12} + 521654265811 p T^{13} + 54633204281 p^{2} T^{14} + 5211503508 p^{3} T^{15} + 873198976 p^{4} T^{16} + 33210707 p^{5} T^{17} + 10802408 p^{6} T^{18} + 119182 p^{7} T^{19} + 90315 p^{8} T^{20} + 173 p^{9} T^{21} + 449 p^{10} T^{22} + p^{12} T^{24} \) | |
| 71 | \( 1 - 19 T + 608 T^{2} - 9307 T^{3} + 179552 T^{4} - 2303574 T^{5} + 33989917 T^{6} - 377566421 T^{7} + 4624247615 T^{8} - 45164214709 T^{9} + 476984599751 T^{10} - 4124585113695 T^{11} + 38252651163656 T^{12} - 4124585113695 p T^{13} + 476984599751 p^{2} T^{14} - 45164214709 p^{3} T^{15} + 4624247615 p^{4} T^{16} - 377566421 p^{5} T^{17} + 33989917 p^{6} T^{18} - 2303574 p^{7} T^{19} + 179552 p^{8} T^{20} - 9307 p^{9} T^{21} + 608 p^{10} T^{22} - 19 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 + 42 T + 1043 T^{2} + 17998 T^{3} + 245747 T^{4} + 2754720 T^{5} + 26880038 T^{6} + 230692403 T^{7} + 1799301587 T^{8} + 12486762683 T^{9} + 80041477558 T^{10} + 485176939647 T^{11} + 3589099987174 T^{12} + 485176939647 p T^{13} + 80041477558 p^{2} T^{14} + 12486762683 p^{3} T^{15} + 1799301587 p^{4} T^{16} + 230692403 p^{5} T^{17} + 26880038 p^{6} T^{18} + 2754720 p^{7} T^{19} + 245747 p^{8} T^{20} + 17998 p^{9} T^{21} + 1043 p^{10} T^{22} + 42 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 + 11 T + 375 T^{2} + 1971 T^{3} + 61812 T^{4} + 141297 T^{5} + 8030226 T^{6} + 5418989 T^{7} + 921051700 T^{8} + 1159310 T^{9} + 93618602276 T^{10} + 97393057 T^{11} + 8156189470360 T^{12} + 97393057 p T^{13} + 93618602276 p^{2} T^{14} + 1159310 p^{3} T^{15} + 921051700 p^{4} T^{16} + 5418989 p^{5} T^{17} + 8030226 p^{6} T^{18} + 141297 p^{7} T^{19} + 61812 p^{8} T^{20} + 1971 p^{9} T^{21} + 375 p^{10} T^{22} + 11 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 56 T + 1951 T^{2} - 49386 T^{3} + 1022271 T^{4} - 17968694 T^{5} + 278807356 T^{6} - 3878361170 T^{7} + 49264947230 T^{8} - 575419961940 T^{9} + 6239757087976 T^{10} - 62947612589627 T^{11} + 593568643259770 T^{12} - 62947612589627 p T^{13} + 6239757087976 p^{2} T^{14} - 575419961940 p^{3} T^{15} + 49264947230 p^{4} T^{16} - 3878361170 p^{5} T^{17} + 278807356 p^{6} T^{18} - 17968694 p^{7} T^{19} + 1022271 p^{8} T^{20} - 49386 p^{9} T^{21} + 1951 p^{10} T^{22} - 56 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 55 T + 2149 T^{2} - 60868 T^{3} + 1441797 T^{4} - 28799664 T^{5} + 507606343 T^{6} - 7923948344 T^{7} + 111878225657 T^{8} - 1429878368183 T^{9} + 16734332409804 T^{10} - 179077113801815 T^{11} + 19821507459342 p T^{12} - 179077113801815 p T^{13} + 16734332409804 p^{2} T^{14} - 1429878368183 p^{3} T^{15} + 111878225657 p^{4} T^{16} - 7923948344 p^{5} T^{17} + 507606343 p^{6} T^{18} - 28799664 p^{7} T^{19} + 1441797 p^{8} T^{20} - 60868 p^{9} T^{21} + 2149 p^{10} T^{22} - 55 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 + 7 T + 483 T^{2} + 3384 T^{3} + 135472 T^{4} + 1005978 T^{5} + 27299194 T^{6} + 202779564 T^{7} + 4278918623 T^{8} + 31268940433 T^{9} + 548132714887 T^{10} + 3783249792977 T^{11} + 58040314464774 T^{12} + 3783249792977 p T^{13} + 548132714887 p^{2} T^{14} + 31268940433 p^{3} T^{15} + 4278918623 p^{4} T^{16} + 202779564 p^{5} T^{17} + 27299194 p^{6} T^{18} + 1005978 p^{7} T^{19} + 135472 p^{8} T^{20} + 3384 p^{9} T^{21} + 483 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.44725836659158218852593435549, −2.40715066173872385388776869684, −2.36455396954994048647127114769, −2.32186860536902330099087141300, −2.31001204397348794065407992430, −2.17748882027530327500197862639, −1.99311848845027408544204974768, −1.85834950938306942531490538669, −1.79862521854351385485678057013, −1.76048347886036180585173721739, −1.75094510066864456827678037203, −1.68798851703583119208542218083, −1.46687867678198896744910628507, −1.42574224192602894671991881084, −1.33572702997548664969784890595, −0.944968550963424123306489681520, −0.902799172959379663664488450822, −0.75842951682753028709877214081, −0.73773271864913604044574985640, −0.70531281681643381679085431585, −0.46463176956490396164089152406, −0.43341153517738273260117183384, −0.37059318581052869958921423254, −0.23351226031614928023110110805, −0.21262736666465604858283523229, 0.21262736666465604858283523229, 0.23351226031614928023110110805, 0.37059318581052869958921423254, 0.43341153517738273260117183384, 0.46463176956490396164089152406, 0.70531281681643381679085431585, 0.73773271864913604044574985640, 0.75842951682753028709877214081, 0.902799172959379663664488450822, 0.944968550963424123306489681520, 1.33572702997548664969784890595, 1.42574224192602894671991881084, 1.46687867678198896744910628507, 1.68798851703583119208542218083, 1.75094510066864456827678037203, 1.76048347886036180585173721739, 1.79862521854351385485678057013, 1.85834950938306942531490538669, 1.99311848845027408544204974768, 2.17748882027530327500197862639, 2.31001204397348794065407992430, 2.32186860536902330099087141300, 2.36455396954994048647127114769, 2.40715066173872385388776869684, 2.44725836659158218852593435549
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.