Dirichlet series
| L(s) = 1 | + 12·3-s + 8·7-s + 78·9-s + 2·11-s + 8·17-s + 10·19-s + 96·21-s + 18·23-s + 364·27-s + 8·29-s − 2·31-s + 24·33-s + 4·37-s + 10·41-s + 28·43-s + 22·47-s + 4·49-s + 96·51-s + 16·53-s + 120·57-s − 2·59-s + 34·61-s + 624·63-s + 32·67-s + 216·69-s + 24·73-s + 16·77-s + ⋯ |
| L(s) = 1 | + 6.92·3-s + 3.02·7-s + 26·9-s + 0.603·11-s + 1.94·17-s + 2.29·19-s + 20.9·21-s + 3.75·23-s + 70.0·27-s + 1.48·29-s − 0.359·31-s + 4.17·33-s + 0.657·37-s + 1.56·41-s + 4.26·43-s + 3.20·47-s + 4/7·49-s + 13.4·51-s + 2.19·53-s + 15.8·57-s − 0.260·59-s + 4.35·61-s + 78.6·63-s + 3.90·67-s + 26.0·69-s + 2.80·73-s + 1.82·77-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 5^{48}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 5^{48}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{24} \cdot 3^{12} \cdot 5^{48}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.12843\times 10^{21}\) |
| Root analytic conductor: | \(7.73872\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{24} \cdot 3^{12} \cdot 5^{48} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(100737.9665\) |
| \(L(\frac12)\) | \(\approx\) | \(100737.9665\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 - T )^{12} \) | |
| 5 | \( 1 \) | |
| good | 7 | \( 1 - 8 T + 60 T^{2} - 316 T^{3} + 1564 T^{4} - 6428 T^{5} + 3627 p T^{6} - 88980 T^{7} + 43334 p T^{8} - 944532 T^{9} + 2876955 T^{10} - 8089000 T^{11} + 22251450 T^{12} - 8089000 p T^{13} + 2876955 p^{2} T^{14} - 944532 p^{3} T^{15} + 43334 p^{5} T^{16} - 88980 p^{5} T^{17} + 3627 p^{7} T^{18} - 6428 p^{7} T^{19} + 1564 p^{8} T^{20} - 316 p^{9} T^{21} + 60 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) |
| 11 | \( 1 - 2 T + 53 T^{2} - 112 T^{3} + 1476 T^{4} - 3362 T^{5} + 29239 T^{6} - 73806 T^{7} + 462344 T^{8} - 1268532 T^{9} + 6190468 T^{10} - 17348506 T^{11} + 72456534 T^{12} - 17348506 p T^{13} + 6190468 p^{2} T^{14} - 1268532 p^{3} T^{15} + 462344 p^{4} T^{16} - 73806 p^{5} T^{17} + 29239 p^{6} T^{18} - 3362 p^{7} T^{19} + 1476 p^{8} T^{20} - 112 p^{9} T^{21} + 53 p^{10} T^{22} - 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 56 T^{2} + 40 T^{3} + 1464 T^{4} + 2680 T^{5} + 25055 T^{6} + 83160 T^{7} + 345840 T^{8} + 1659920 T^{9} + 4458576 T^{10} + 25355440 T^{11} + 57470441 T^{12} + 25355440 p T^{13} + 4458576 p^{2} T^{14} + 1659920 p^{3} T^{15} + 345840 p^{4} T^{16} + 83160 p^{5} T^{17} + 25055 p^{6} T^{18} + 2680 p^{7} T^{19} + 1464 p^{8} T^{20} + 40 p^{9} T^{21} + 56 p^{10} T^{22} + p^{12} T^{24} \) | |
| 17 | \( 1 - 8 T + 105 T^{2} - 806 T^{3} + 6539 T^{4} - 41918 T^{5} + 276234 T^{6} - 1530530 T^{7} + 8535883 T^{8} - 42118202 T^{9} + 205227095 T^{10} - 900374290 T^{11} + 3914787535 T^{12} - 900374290 p T^{13} + 205227095 p^{2} T^{14} - 42118202 p^{3} T^{15} + 8535883 p^{4} T^{16} - 1530530 p^{5} T^{17} + 276234 p^{6} T^{18} - 41918 p^{7} T^{19} + 6539 p^{8} T^{20} - 806 p^{9} T^{21} + 105 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 - 10 T + 143 T^{2} - 1190 T^{3} + 559 p T^{4} - 73450 T^{5} + 512570 T^{6} - 3067700 T^{7} + 18004530 T^{8} - 5000550 p T^{9} + 486474863 T^{10} - 2284834980 T^{11} + 10364053264 T^{12} - 2284834980 p T^{13} + 486474863 p^{2} T^{14} - 5000550 p^{4} T^{15} + 18004530 p^{4} T^{16} - 3067700 p^{5} T^{17} + 512570 p^{6} T^{18} - 73450 p^{7} T^{19} + 559 p^{9} T^{20} - 1190 p^{9} T^{21} + 143 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 18 T + 287 T^{2} - 3024 T^{3} + 29704 T^{4} - 237698 T^{5} + 1824489 T^{6} - 12237956 T^{7} + 79502112 T^{8} - 464127842 T^{9} + 2631388012 T^{10} - 13575471026 T^{11} + 68116560214 T^{12} - 13575471026 p T^{13} + 2631388012 p^{2} T^{14} - 464127842 p^{3} T^{15} + 79502112 p^{4} T^{16} - 12237956 p^{5} T^{17} + 1824489 p^{6} T^{18} - 237698 p^{7} T^{19} + 29704 p^{8} T^{20} - 3024 p^{9} T^{21} + 287 p^{10} T^{22} - 18 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 8 T + 209 T^{2} - 1602 T^{3} + 22351 T^{4} - 161548 T^{5} + 1600884 T^{6} - 10744102 T^{7} + 84736641 T^{8} - 522133958 T^{9} + 3473462939 T^{10} - 19418833262 T^{11} + 112780838371 T^{12} - 19418833262 p T^{13} + 3473462939 p^{2} T^{14} - 522133958 p^{3} T^{15} + 84736641 p^{4} T^{16} - 10744102 p^{5} T^{17} + 1600884 p^{6} T^{18} - 161548 p^{7} T^{19} + 22351 p^{8} T^{20} - 1602 p^{9} T^{21} + 209 p^{10} T^{22} - 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 2 T + 173 T^{2} + 642 T^{3} + 15426 T^{4} + 74812 T^{5} + 971229 T^{6} + 5093456 T^{7} + 48544804 T^{8} + 243129922 T^{9} + 2002481678 T^{10} + 9061479886 T^{11} + 68362715154 T^{12} + 9061479886 p T^{13} + 2002481678 p^{2} T^{14} + 243129922 p^{3} T^{15} + 48544804 p^{4} T^{16} + 5093456 p^{5} T^{17} + 971229 p^{6} T^{18} + 74812 p^{7} T^{19} + 15426 p^{8} T^{20} + 642 p^{9} T^{21} + 173 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 - 4 T + 123 T^{2} - 348 T^{3} + 8319 T^{4} - 26284 T^{5} + 435886 T^{6} - 1651932 T^{7} + 16772207 T^{8} - 76756396 T^{9} + 550829103 T^{10} - 3394858512 T^{11} + 19811224119 T^{12} - 3394858512 p T^{13} + 550829103 p^{2} T^{14} - 76756396 p^{3} T^{15} + 16772207 p^{4} T^{16} - 1651932 p^{5} T^{17} + 435886 p^{6} T^{18} - 26284 p^{7} T^{19} + 8319 p^{8} T^{20} - 348 p^{9} T^{21} + 123 p^{10} T^{22} - 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( 1 - 10 T + 7 p T^{2} - 2560 T^{3} + 41491 T^{4} - 334200 T^{5} + 4000105 T^{6} - 717550 p T^{7} + 287099130 T^{8} - 1933117950 T^{9} + 16181194587 T^{10} - 99324777420 T^{11} + 735586681699 T^{12} - 99324777420 p T^{13} + 16181194587 p^{2} T^{14} - 1933117950 p^{3} T^{15} + 287099130 p^{4} T^{16} - 717550 p^{6} T^{17} + 4000105 p^{6} T^{18} - 334200 p^{7} T^{19} + 41491 p^{8} T^{20} - 2560 p^{9} T^{21} + 7 p^{11} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 28 T + 14 p T^{2} - 9124 T^{3} + 120669 T^{4} - 1331548 T^{5} + 13443264 T^{6} - 120385296 T^{7} + 1015717232 T^{8} - 7855805732 T^{9} + 58429265502 T^{10} - 406098986136 T^{11} + 2745698906324 T^{12} - 406098986136 p T^{13} + 58429265502 p^{2} T^{14} - 7855805732 p^{3} T^{15} + 1015717232 p^{4} T^{16} - 120385296 p^{5} T^{17} + 13443264 p^{6} T^{18} - 1331548 p^{7} T^{19} + 120669 p^{8} T^{20} - 9124 p^{9} T^{21} + 14 p^{11} T^{22} - 28 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 - 22 T + 590 T^{2} - 8794 T^{3} + 140594 T^{4} - 1624772 T^{5} + 19546119 T^{6} - 186775380 T^{7} + 1844204078 T^{8} - 15113438958 T^{9} + 127945132115 T^{10} - 918180204890 T^{11} + 6814783999550 T^{12} - 918180204890 p T^{13} + 127945132115 p^{2} T^{14} - 15113438958 p^{3} T^{15} + 1844204078 p^{4} T^{16} - 186775380 p^{5} T^{17} + 19546119 p^{6} T^{18} - 1624772 p^{7} T^{19} + 140594 p^{8} T^{20} - 8794 p^{9} T^{21} + 590 p^{10} T^{22} - 22 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 - 16 T + 365 T^{2} - 4018 T^{3} + 54659 T^{4} - 439206 T^{5} + 4454566 T^{6} - 25497930 T^{7} + 215950183 T^{8} - 13409118 p T^{9} + 6490400795 T^{10} + 136618250 T^{11} + 199822350295 T^{12} + 136618250 p T^{13} + 6490400795 p^{2} T^{14} - 13409118 p^{4} T^{15} + 215950183 p^{4} T^{16} - 25497930 p^{5} T^{17} + 4454566 p^{6} T^{18} - 439206 p^{7} T^{19} + 54659 p^{8} T^{20} - 4018 p^{9} T^{21} + 365 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 2 T + 299 T^{2} + 968 T^{3} + 48991 T^{4} + 175602 T^{5} + 5841294 T^{6} + 20890728 T^{7} + 548065746 T^{8} + 1912281622 T^{9} + 42299596739 T^{10} + 138931513138 T^{11} + 2730090656756 T^{12} + 138931513138 p T^{13} + 42299596739 p^{2} T^{14} + 1912281622 p^{3} T^{15} + 548065746 p^{4} T^{16} + 20890728 p^{5} T^{17} + 5841294 p^{6} T^{18} + 175602 p^{7} T^{19} + 48991 p^{8} T^{20} + 968 p^{9} T^{21} + 299 p^{10} T^{22} + 2 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 - 34 T + 931 T^{2} - 17784 T^{3} + 299771 T^{4} - 4219904 T^{5} + 54478221 T^{6} - 623523274 T^{7} + 6675819926 T^{8} - 65131415834 T^{9} + 601255524031 T^{10} - 5126109462784 T^{11} + 41575780979911 T^{12} - 5126109462784 p T^{13} + 601255524031 p^{2} T^{14} - 65131415834 p^{3} T^{15} + 6675819926 p^{4} T^{16} - 623523274 p^{5} T^{17} + 54478221 p^{6} T^{18} - 4219904 p^{7} T^{19} + 299771 p^{8} T^{20} - 17784 p^{9} T^{21} + 931 p^{10} T^{22} - 34 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 32 T + 930 T^{2} - 18284 T^{3} + 331129 T^{4} - 4942272 T^{5} + 68966444 T^{6} - 844689360 T^{7} + 9754232188 T^{8} - 101576469648 T^{9} + 1002214713130 T^{10} - 9026621184500 T^{11} + 77198209555300 T^{12} - 9026621184500 p T^{13} + 1002214713130 p^{2} T^{14} - 101576469648 p^{3} T^{15} + 9754232188 p^{4} T^{16} - 844689360 p^{5} T^{17} + 68966444 p^{6} T^{18} - 4942272 p^{7} T^{19} + 331129 p^{8} T^{20} - 18284 p^{9} T^{21} + 930 p^{10} T^{22} - 32 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 342 T^{2} - 290 T^{3} + 66046 T^{4} - 89060 T^{5} + 9213265 T^{6} - 15439940 T^{7} + 1024937090 T^{8} - 1797911460 T^{9} + 94544477857 T^{10} - 159539967490 T^{11} + 7313789156734 T^{12} - 159539967490 p T^{13} + 94544477857 p^{2} T^{14} - 1797911460 p^{3} T^{15} + 1024937090 p^{4} T^{16} - 15439940 p^{5} T^{17} + 9213265 p^{6} T^{18} - 89060 p^{7} T^{19} + 66046 p^{8} T^{20} - 290 p^{9} T^{21} + 342 p^{10} T^{22} + p^{12} T^{24} \) | |
| 73 | \( 1 - 24 T + 685 T^{2} - 11512 T^{3} + 200414 T^{4} - 2688544 T^{5} + 36156166 T^{6} - 412862400 T^{7} + 4682083863 T^{8} - 47168353696 T^{9} + 470410990190 T^{10} - 4256249367760 T^{11} + 38063170259995 T^{12} - 4256249367760 p T^{13} + 470410990190 p^{2} T^{14} - 47168353696 p^{3} T^{15} + 4682083863 p^{4} T^{16} - 412862400 p^{5} T^{17} + 36156166 p^{6} T^{18} - 2688544 p^{7} T^{19} + 200414 p^{8} T^{20} - 11512 p^{9} T^{21} + 685 p^{10} T^{22} - 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 6 T + 537 T^{2} - 4234 T^{3} + 153666 T^{4} - 1275456 T^{5} + 30367081 T^{6} - 242347292 T^{7} + 4442092304 T^{8} - 33097177246 T^{9} + 501208175882 T^{10} - 3398002920062 T^{11} + 44539225232194 T^{12} - 3398002920062 p T^{13} + 501208175882 p^{2} T^{14} - 33097177246 p^{3} T^{15} + 4442092304 p^{4} T^{16} - 242347292 p^{5} T^{17} + 30367081 p^{6} T^{18} - 1275456 p^{7} T^{19} + 153666 p^{8} T^{20} - 4234 p^{9} T^{21} + 537 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 - 28 T + 1052 T^{2} - 21444 T^{3} + 477224 T^{4} - 7727928 T^{5} + 128649444 T^{6} - 1730017936 T^{7} + 23378831032 T^{8} - 267418884132 T^{9} + 3045189399652 T^{10} - 29981846916356 T^{11} + 292845072708534 T^{12} - 29981846916356 p T^{13} + 3045189399652 p^{2} T^{14} - 267418884132 p^{3} T^{15} + 23378831032 p^{4} T^{16} - 1730017936 p^{5} T^{17} + 128649444 p^{6} T^{18} - 7727928 p^{7} T^{19} + 477224 p^{8} T^{20} - 21444 p^{9} T^{21} + 1052 p^{10} T^{22} - 28 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 - 10 T + 588 T^{2} - 5680 T^{3} + 174686 T^{4} - 1500640 T^{5} + 33804555 T^{6} - 251645690 T^{7} + 4765318420 T^{8} - 30843043440 T^{9} + 533214292458 T^{10} - 3088687785120 T^{11} + 50728438283049 T^{12} - 3088687785120 p T^{13} + 533214292458 p^{2} T^{14} - 30843043440 p^{3} T^{15} + 4765318420 p^{4} T^{16} - 251645690 p^{5} T^{17} + 33804555 p^{6} T^{18} - 1500640 p^{7} T^{19} + 174686 p^{8} T^{20} - 5680 p^{9} T^{21} + 588 p^{10} T^{22} - 10 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 16 T + 643 T^{2} - 7792 T^{3} + 185439 T^{4} - 1762696 T^{5} + 32597681 T^{6} - 242946928 T^{7} + 4060867002 T^{8} - 23625077704 T^{9} + 411243311383 T^{10} - 1986049594088 T^{11} + 39405273192799 T^{12} - 1986049594088 p T^{13} + 411243311383 p^{2} T^{14} - 23625077704 p^{3} T^{15} + 4060867002 p^{4} T^{16} - 242946928 p^{5} T^{17} + 32597681 p^{6} T^{18} - 1762696 p^{7} T^{19} + 185439 p^{8} T^{20} - 7792 p^{9} T^{21} + 643 p^{10} T^{22} - 16 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−2.42354834683020272435061069627, −2.08180986796550988841568673987, −2.04613697108456957996472743505, −2.03902433819952127039708653118, −1.99094026514467808026147456972, −1.97076871985965162129799159134, −1.85300344574785426663291488849, −1.85196115175857907189689453849, −1.83225666429736404406630047377, −1.82697005222597703442515736068, −1.74912414204812923917921364583, −1.60802996945665613224507406833, −1.51731785227298098594008721923, −1.13353965148705270727676309968, −1.00570188032626464484649195028, −0.989569165714878291437276167750, −0.952713141115918096066760169945, −0.888997884612696181141123877305, −0.888981798532948323438824504677, −0.864211006226909868984360382922, −0.849336603561742906430404392882, −0.803090753023393581601096676886, −0.60866909801663063748933718146, −0.53857581947606624304283391254, −0.52846444007236409334725216849, 0.52846444007236409334725216849, 0.53857581947606624304283391254, 0.60866909801663063748933718146, 0.803090753023393581601096676886, 0.849336603561742906430404392882, 0.864211006226909868984360382922, 0.888981798532948323438824504677, 0.888997884612696181141123877305, 0.952713141115918096066760169945, 0.989569165714878291437276167750, 1.00570188032626464484649195028, 1.13353965148705270727676309968, 1.51731785227298098594008721923, 1.60802996945665613224507406833, 1.74912414204812923917921364583, 1.82697005222597703442515736068, 1.83225666429736404406630047377, 1.85196115175857907189689453849, 1.85300344574785426663291488849, 1.97076871985965162129799159134, 1.99094026514467808026147456972, 2.03902433819952127039708653118, 2.04613697108456957996472743505, 2.08180986796550988841568673987, 2.42354834683020272435061069627
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.