Dirichlet series
| L(s) = 1 | + 9·5-s + 6·7-s + 2·8-s + 3·11-s − 9·13-s + 15·17-s + 9·19-s + 3·23-s + 42·25-s − 2·27-s + 3·29-s + 6·31-s + 54·35-s + 18·40-s − 27·41-s + 6·43-s − 6·47-s + 15·49-s − 9·53-s + 27·55-s + 12·56-s − 24·59-s − 15·61-s + 64-s − 81·65-s + 33·67-s − 33·71-s + ⋯ |
| L(s) = 1 | + 4.02·5-s + 2.26·7-s + 0.707·8-s + 0.904·11-s − 2.49·13-s + 3.63·17-s + 2.06·19-s + 0.625·23-s + 42/5·25-s − 0.384·27-s + 0.557·29-s + 1.07·31-s + 9.12·35-s + 2.84·40-s − 4.21·41-s + 0.914·43-s − 0.875·47-s + 15/7·49-s − 1.23·53-s + 3.64·55-s + 1.60·56-s − 3.12·59-s − 1.92·61-s + 1/8·64-s − 10.0·65-s + 4.03·67-s − 3.91·71-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 7^{12} \cdot 19^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{12} \cdot 7^{12} \cdot 19^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(4.48084\times 10^{9}\) |
| Root analytic conductor: | \(2.52429\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{12} \cdot 7^{12} \cdot 19^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(121.2861816\) |
| \(L(\frac12)\) | \(\approx\) | \(121.2861816\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T^{3} + T^{6} )^{2} \) |
| 3 | \( ( 1 + T^{3} + T^{6} )^{2} \) | |
| 7 | \( ( 1 - T + T^{2} )^{6} \) | |
| 19 | \( 1 - 9 T + 54 T^{2} - 288 T^{3} + 1530 T^{4} - 8667 T^{5} + 40789 T^{6} - 8667 p T^{7} + 1530 p^{2} T^{8} - 288 p^{3} T^{9} + 54 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 5 | \( 1 - 9 T + 39 T^{2} - 108 T^{3} + 9 p^{2} T^{4} - 504 T^{5} + 1533 T^{6} - 4608 T^{7} + 10998 T^{8} - 22896 T^{9} + 11166 p T^{10} - 31833 p T^{11} + 400661 T^{12} - 31833 p^{2} T^{13} + 11166 p^{3} T^{14} - 22896 p^{3} T^{15} + 10998 p^{4} T^{16} - 4608 p^{5} T^{17} + 1533 p^{6} T^{18} - 504 p^{7} T^{19} + 9 p^{10} T^{20} - 108 p^{9} T^{21} + 39 p^{10} T^{22} - 9 p^{11} T^{23} + p^{12} T^{24} \) |
| 11 | \( 1 - 3 T - 30 T^{2} - 23 T^{3} + 789 T^{4} + 1644 T^{5} - 7929 T^{6} - 42831 T^{7} + 23280 T^{8} + 456747 T^{9} + 911907 T^{10} - 2413332 T^{11} - 13589831 T^{12} - 2413332 p T^{13} + 911907 p^{2} T^{14} + 456747 p^{3} T^{15} + 23280 p^{4} T^{16} - 42831 p^{5} T^{17} - 7929 p^{6} T^{18} + 1644 p^{7} T^{19} + 789 p^{8} T^{20} - 23 p^{9} T^{21} - 30 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 13 | \( 1 + 9 T + 63 T^{2} + 346 T^{3} + 1944 T^{4} + 774 p T^{5} + 49762 T^{6} + 222444 T^{7} + 978597 T^{8} + 4068907 T^{9} + 16480017 T^{10} + 62146620 T^{11} + 228854089 T^{12} + 62146620 p T^{13} + 16480017 p^{2} T^{14} + 4068907 p^{3} T^{15} + 978597 p^{4} T^{16} + 222444 p^{5} T^{17} + 49762 p^{6} T^{18} + 774 p^{8} T^{19} + 1944 p^{8} T^{20} + 346 p^{9} T^{21} + 63 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 - 15 T + 141 T^{2} - 1058 T^{3} + 6771 T^{4} - 2325 p T^{5} + 216851 T^{6} - 1133370 T^{7} + 5653296 T^{8} - 26981030 T^{9} + 123872448 T^{10} - 545291670 T^{11} + 2295782281 T^{12} - 545291670 p T^{13} + 123872448 p^{2} T^{14} - 26981030 p^{3} T^{15} + 5653296 p^{4} T^{16} - 1133370 p^{5} T^{17} + 216851 p^{6} T^{18} - 2325 p^{8} T^{19} + 6771 p^{8} T^{20} - 1058 p^{9} T^{21} + 141 p^{10} T^{22} - 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( 1 - 3 T - 30 T^{2} + 233 T^{3} + 678 T^{4} - 8403 T^{5} - 2823 T^{6} + 254049 T^{7} - 493587 T^{8} - 4628124 T^{9} + 24584292 T^{10} + 45906228 T^{11} - 651404435 T^{12} + 45906228 p T^{13} + 24584292 p^{2} T^{14} - 4628124 p^{3} T^{15} - 493587 p^{4} T^{16} + 254049 p^{5} T^{17} - 2823 p^{6} T^{18} - 8403 p^{7} T^{19} + 678 p^{8} T^{20} + 233 p^{9} T^{21} - 30 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 29 | \( 1 - 3 T - 12 T^{2} + 134 T^{3} - 2463 T^{4} + 10173 T^{5} + 11037 T^{6} - 394608 T^{7} + 3946041 T^{8} - 13804497 T^{9} - 388539 T^{10} + 454492827 T^{11} - 4295034617 T^{12} + 454492827 p T^{13} - 388539 p^{2} T^{14} - 13804497 p^{3} T^{15} + 3946041 p^{4} T^{16} - 394608 p^{5} T^{17} + 11037 p^{6} T^{18} + 10173 p^{7} T^{19} - 2463 p^{8} T^{20} + 134 p^{9} T^{21} - 12 p^{10} T^{22} - 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 - 6 T - 102 T^{2} + 636 T^{3} + 5385 T^{4} - 30495 T^{5} - 257422 T^{6} + 994797 T^{7} + 11976660 T^{8} - 26731797 T^{9} - 449473788 T^{10} + 360747045 T^{11} + 14360706483 T^{12} + 360747045 p T^{13} - 449473788 p^{2} T^{14} - 26731797 p^{3} T^{15} + 11976660 p^{4} T^{16} + 994797 p^{5} T^{17} - 257422 p^{6} T^{18} - 30495 p^{7} T^{19} + 5385 p^{8} T^{20} + 636 p^{9} T^{21} - 102 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( ( 1 + 114 T^{2} - 425 T^{3} + 5643 T^{4} - 39723 T^{5} + 215447 T^{6} - 39723 p T^{7} + 5643 p^{2} T^{8} - 425 p^{3} T^{9} + 114 p^{4} T^{10} + p^{6} T^{12} )^{2} \) | |
| 41 | \( 1 + 27 T + 279 T^{2} + 1215 T^{3} + 1773 T^{4} + 13320 T^{5} + 62255 T^{6} - 1473426 T^{7} - 11603682 T^{8} + 39792465 T^{9} + 886282614 T^{10} + 5443368489 T^{11} + 28254084501 T^{12} + 5443368489 p T^{13} + 886282614 p^{2} T^{14} + 39792465 p^{3} T^{15} - 11603682 p^{4} T^{16} - 1473426 p^{5} T^{17} + 62255 p^{6} T^{18} + 13320 p^{7} T^{19} + 1773 p^{8} T^{20} + 1215 p^{9} T^{21} + 279 p^{10} T^{22} + 27 p^{11} T^{23} + p^{12} T^{24} \) | |
| 43 | \( 1 - 6 T + 186 T^{2} - 919 T^{3} + 15768 T^{4} - 50598 T^{5} + 696356 T^{6} + 146616 T^{7} + 9936192 T^{8} + 197933067 T^{9} - 720858576 T^{10} + 14903560038 T^{11} - 52853784955 T^{12} + 14903560038 p T^{13} - 720858576 p^{2} T^{14} + 197933067 p^{3} T^{15} + 9936192 p^{4} T^{16} + 146616 p^{5} T^{17} + 696356 p^{6} T^{18} - 50598 p^{7} T^{19} + 15768 p^{8} T^{20} - 919 p^{9} T^{21} + 186 p^{10} T^{22} - 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 47 | \( 1 + 6 T + 57 T^{2} - 218 T^{3} + 942 T^{4} - 240 T^{5} + 209004 T^{6} + 27501 T^{7} + 1235880 T^{8} - 100003527 T^{9} - 148365828 T^{10} - 2314505988 T^{11} + 21194445661 T^{12} - 2314505988 p T^{13} - 148365828 p^{2} T^{14} - 100003527 p^{3} T^{15} + 1235880 p^{4} T^{16} + 27501 p^{5} T^{17} + 209004 p^{6} T^{18} - 240 p^{7} T^{19} + 942 p^{8} T^{20} - 218 p^{9} T^{21} + 57 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( 1 + 9 T - 21 T^{2} - 1079 T^{3} - 8514 T^{4} + 15960 T^{5} + 769575 T^{6} + 4771287 T^{7} - 19260579 T^{8} - 444121290 T^{9} - 1916617143 T^{10} + 12417235827 T^{11} + 202347197779 T^{12} + 12417235827 p T^{13} - 1916617143 p^{2} T^{14} - 444121290 p^{3} T^{15} - 19260579 p^{4} T^{16} + 4771287 p^{5} T^{17} + 769575 p^{6} T^{18} + 15960 p^{7} T^{19} - 8514 p^{8} T^{20} - 1079 p^{9} T^{21} - 21 p^{10} T^{22} + 9 p^{11} T^{23} + p^{12} T^{24} \) | |
| 59 | \( 1 + 24 T + 192 T^{2} + 425 T^{3} + 60 p T^{4} + 88452 T^{5} + 425112 T^{6} - 2122140 T^{7} - 15531222 T^{8} + 14175315 T^{9} - 449660352 T^{10} - 5741083770 T^{11} - 23061745019 T^{12} - 5741083770 p T^{13} - 449660352 p^{2} T^{14} + 14175315 p^{3} T^{15} - 15531222 p^{4} T^{16} - 2122140 p^{5} T^{17} + 425112 p^{6} T^{18} + 88452 p^{7} T^{19} + 60 p^{9} T^{20} + 425 p^{9} T^{21} + 192 p^{10} T^{22} + 24 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( 1 + 15 T + 129 T^{2} + 1191 T^{3} + 15315 T^{4} + 111261 T^{5} + 583307 T^{6} + 2497743 T^{7} + 3618639 T^{8} - 246935550 T^{9} - 3398786712 T^{10} - 35046876150 T^{11} - 285450746949 T^{12} - 35046876150 p T^{13} - 3398786712 p^{2} T^{14} - 246935550 p^{3} T^{15} + 3618639 p^{4} T^{16} + 2497743 p^{5} T^{17} + 583307 p^{6} T^{18} + 111261 p^{7} T^{19} + 15315 p^{8} T^{20} + 1191 p^{9} T^{21} + 129 p^{10} T^{22} + 15 p^{11} T^{23} + p^{12} T^{24} \) | |
| 67 | \( 1 - 33 T + 570 T^{2} - 6442 T^{3} + 64020 T^{4} - 711582 T^{5} + 8396857 T^{6} - 85989132 T^{7} + 765030375 T^{8} - 6567231427 T^{9} + 59520481296 T^{10} - 538988942694 T^{11} + 4611115659739 T^{12} - 538988942694 p T^{13} + 59520481296 p^{2} T^{14} - 6567231427 p^{3} T^{15} + 765030375 p^{4} T^{16} - 85989132 p^{5} T^{17} + 8396857 p^{6} T^{18} - 711582 p^{7} T^{19} + 64020 p^{8} T^{20} - 6442 p^{9} T^{21} + 570 p^{10} T^{22} - 33 p^{11} T^{23} + p^{12} T^{24} \) | |
| 71 | \( 1 + 33 T + 795 T^{2} + 13894 T^{3} + 207879 T^{4} + 2733513 T^{5} + 32833355 T^{6} + 366281388 T^{7} + 3817767888 T^{8} + 37675300390 T^{9} + 353765315220 T^{10} + 3179133007302 T^{11} + 27382233363853 T^{12} + 3179133007302 p T^{13} + 353765315220 p^{2} T^{14} + 37675300390 p^{3} T^{15} + 3817767888 p^{4} T^{16} + 366281388 p^{5} T^{17} + 32833355 p^{6} T^{18} + 2733513 p^{7} T^{19} + 207879 p^{8} T^{20} + 13894 p^{9} T^{21} + 795 p^{10} T^{22} + 33 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( 1 - 33 T + 480 T^{2} - 4298 T^{3} + 32076 T^{4} - 228096 T^{5} + 918761 T^{6} + 5018496 T^{7} - 121959831 T^{8} + 1541203731 T^{9} - 16225783224 T^{10} + 128466512886 T^{11} - 956499862441 T^{12} + 128466512886 p T^{13} - 16225783224 p^{2} T^{14} + 1541203731 p^{3} T^{15} - 121959831 p^{4} T^{16} + 5018496 p^{5} T^{17} + 918761 p^{6} T^{18} - 228096 p^{7} T^{19} + 32076 p^{8} T^{20} - 4298 p^{9} T^{21} + 480 p^{10} T^{22} - 33 p^{11} T^{23} + p^{12} T^{24} \) | |
| 79 | \( 1 - 33 T + 549 T^{2} - 6438 T^{3} + 67104 T^{4} - 604902 T^{5} + 3957104 T^{6} - 9365103 T^{7} - 163899918 T^{8} + 3126729324 T^{9} - 37807435161 T^{10} + 423546920334 T^{11} - 4141547152575 T^{12} + 423546920334 p T^{13} - 37807435161 p^{2} T^{14} + 3126729324 p^{3} T^{15} - 163899918 p^{4} T^{16} - 9365103 p^{5} T^{17} + 3957104 p^{6} T^{18} - 604902 p^{7} T^{19} + 67104 p^{8} T^{20} - 6438 p^{9} T^{21} + 549 p^{10} T^{22} - 33 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 36 T + 375 T^{2} + 1062 T^{3} + 37668 T^{4} + 709299 T^{5} + 106646 T^{6} - 32309793 T^{7} + 507950397 T^{8} + 4172487768 T^{9} - 45520667766 T^{10} + 1080537948 T^{11} + 6418649781519 T^{12} + 1080537948 p T^{13} - 45520667766 p^{2} T^{14} + 4172487768 p^{3} T^{15} + 507950397 p^{4} T^{16} - 32309793 p^{5} T^{17} + 106646 p^{6} T^{18} + 709299 p^{7} T^{19} + 37668 p^{8} T^{20} + 1062 p^{9} T^{21} + 375 p^{10} T^{22} + 36 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 6 T + 333 T^{2} + 1507 T^{3} + 55383 T^{4} + 243942 T^{5} + 7061337 T^{6} + 44798019 T^{7} + 790586016 T^{8} + 6591256776 T^{9} + 74031767277 T^{10} + 706796386224 T^{11} + 6389036003719 T^{12} + 706796386224 p T^{13} + 74031767277 p^{2} T^{14} + 6591256776 p^{3} T^{15} + 790586016 p^{4} T^{16} + 44798019 p^{5} T^{17} + 7061337 p^{6} T^{18} + 243942 p^{7} T^{19} + 55383 p^{8} T^{20} + 1507 p^{9} T^{21} + 333 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( 1 - 21 T + 27 T^{2} + 942 T^{3} + 30240 T^{4} - 498468 T^{5} + 404206 T^{6} + 17715843 T^{7} + 235851156 T^{8} - 3782055726 T^{9} + 2085795819 T^{10} + 30544541712 T^{11} + 1262932812723 T^{12} + 30544541712 p T^{13} + 2085795819 p^{2} T^{14} - 3782055726 p^{3} T^{15} + 235851156 p^{4} T^{16} + 17715843 p^{5} T^{17} + 404206 p^{6} T^{18} - 498468 p^{7} T^{19} + 30240 p^{8} T^{20} + 942 p^{9} T^{21} + 27 p^{10} T^{22} - 21 p^{11} T^{23} + p^{12} T^{24} \) | |
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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
−3.24483403545996509378916969010, −3.13324481898389878728177337968, −3.04872331418209218907659558950, −3.04347152659115455500294168701, −2.91001740061426396100074989839, −2.82711274303087818946465753003, −2.81832060972115242507743959703, −2.51376372405288516648837859731, −2.45383503021175679187781393000, −2.40551245942716618858971373320, −2.15124955950530800872923790352, −2.04173344485974203753826187347, −1.85090277205121043153470659104, −1.83576528023637968246730073294, −1.83018390326124953281583972717, −1.78531770863073467352179111412, −1.67735094248997761987660463910, −1.41246467825636539070736245336, −1.36666877765084171643025537436, −1.36026681507243532751559719060, −1.06047880379250540714604902550, −1.02492015474859287418235629492, −0.61742609076097742054919070479, −0.58807015000200133213144905383, −0.55218245797206579071017820041, 0.55218245797206579071017820041, 0.58807015000200133213144905383, 0.61742609076097742054919070479, 1.02492015474859287418235629492, 1.06047880379250540714604902550, 1.36026681507243532751559719060, 1.36666877765084171643025537436, 1.41246467825636539070736245336, 1.67735094248997761987660463910, 1.78531770863073467352179111412, 1.83018390326124953281583972717, 1.83576528023637968246730073294, 1.85090277205121043153470659104, 2.04173344485974203753826187347, 2.15124955950530800872923790352, 2.40551245942716618858971373320, 2.45383503021175679187781393000, 2.51376372405288516648837859731, 2.81832060972115242507743959703, 2.82711274303087818946465753003, 2.91001740061426396100074989839, 3.04347152659115455500294168701, 3.04872331418209218907659558950, 3.13324481898389878728177337968, 3.24483403545996509378916969010
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.