Dirichlet series
| L(s) = 1 | + 3·5-s − 3·7-s − 18·11-s − 5·13-s − 6·17-s − 81·23-s + 43·25-s + 69·29-s − 45·31-s − 9·35-s + 20·37-s − 54·41-s + 207·47-s − 73·49-s − 252·53-s − 54·55-s + 306·59-s − 7·61-s − 15·65-s + 12·67-s + 74·73-s + 54·77-s − 33·79-s − 549·83-s − 18·85-s + 168·89-s + 15·91-s + ⋯ |
| L(s) = 1 | + 3/5·5-s − 3/7·7-s − 1.63·11-s − 0.384·13-s − 0.352·17-s − 3.52·23-s + 1.71·25-s + 2.37·29-s − 1.45·31-s − 0.257·35-s + 0.540·37-s − 1.31·41-s + 4.40·47-s − 1.48·49-s − 4.75·53-s − 0.981·55-s + 5.18·59-s − 0.114·61-s − 0.230·65-s + 0.179·67-s + 1.01·73-s + 0.701·77-s − 0.417·79-s − 6.61·83-s − 0.211·85-s + 1.88·89-s + 0.164·91-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{48} \cdot 3^{24}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.41562\times 10^{13}\) |
| Root analytic conductor: | \(6.86182\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{48} \cdot 3^{24} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(1.959780168\) |
| \(L(\frac12)\) | \(\approx\) | \(1.959780168\) |
| \(L(2)\) | 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 | \( 1 \) | |
| good | 5 | \( 1 - 3 T - 34 T^{2} + 339 T^{3} + 211 T^{4} - 10296 T^{5} + 50708 T^{6} + 156576 T^{7} - 1757804 T^{8} + 156576 p^{2} T^{9} + 50708 p^{4} T^{10} - 10296 p^{6} T^{11} + 211 p^{8} T^{12} + 339 p^{10} T^{13} - 34 p^{12} T^{14} - 3 p^{14} T^{15} + p^{16} T^{16} \) |
| 7 | \( 1 + 3 T + 82 T^{2} + 237 T^{3} + 2419 T^{4} + 21600 T^{5} + 116980 T^{6} + 2348256 T^{7} + 7641628 T^{8} + 2348256 p^{2} T^{9} + 116980 p^{4} T^{10} + 21600 p^{6} T^{11} + 2419 p^{8} T^{12} + 237 p^{10} T^{13} + 82 p^{12} T^{14} + 3 p^{14} T^{15} + p^{16} T^{16} \) | |
| 11 | \( 1 + 18 T + 448 T^{2} + 6120 T^{3} + 93463 T^{4} + 1253556 T^{5} + 16013428 T^{6} + 201806226 T^{7} + 2222076136 T^{8} + 201806226 p^{2} T^{9} + 16013428 p^{4} T^{10} + 1253556 p^{6} T^{11} + 93463 p^{8} T^{12} + 6120 p^{10} T^{13} + 448 p^{12} T^{14} + 18 p^{14} T^{15} + p^{16} T^{16} \) | |
| 13 | \( 1 + 5 T - 360 T^{2} + 1561 T^{3} + 77501 T^{4} - 597312 T^{5} - 5007086 T^{6} + 5241266 p T^{7} + 60552144 T^{8} + 5241266 p^{3} T^{9} - 5007086 p^{4} T^{10} - 597312 p^{6} T^{11} + 77501 p^{8} T^{12} + 1561 p^{10} T^{13} - 360 p^{12} T^{14} + 5 p^{14} T^{15} + p^{16} T^{16} \) | |
| 17 | \( ( 1 + 3 T + 334 T^{2} + 693 T^{3} + 110178 T^{4} + 693 p^{2} T^{5} + 334 p^{4} T^{6} + 3 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 19 | \( 1 - 1157 T^{2} + 639226 T^{4} - 232952123 T^{6} + 79410905002 T^{8} - 232952123 p^{4} T^{10} + 639226 p^{8} T^{12} - 1157 p^{12} T^{14} + p^{16} T^{16} \) | |
| 23 | \( 1 + 81 T + 4870 T^{2} + 217323 T^{3} + 8414383 T^{4} + 278460504 T^{5} + 8325324592 T^{6} + 221238967356 T^{7} + 5380294547716 T^{8} + 221238967356 p^{2} T^{9} + 8325324592 p^{4} T^{10} + 278460504 p^{6} T^{11} + 8414383 p^{8} T^{12} + 217323 p^{10} T^{13} + 4870 p^{12} T^{14} + 81 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( 1 - 69 T + 1976 T^{2} - 82905 T^{3} + 2728573 T^{4} - 24459264 T^{5} + 249830930 T^{6} + 10215187926 T^{7} - 1249058812400 T^{8} + 10215187926 p^{2} T^{9} + 249830930 p^{4} T^{10} - 24459264 p^{6} T^{11} + 2728573 p^{8} T^{12} - 82905 p^{10} T^{13} + 1976 p^{12} T^{14} - 69 p^{14} T^{15} + p^{16} T^{16} \) | |
| 31 | \( 1 + 45 T + 3520 T^{2} + 128025 T^{3} + 6239425 T^{4} + 221987952 T^{5} + 7979134318 T^{6} + 268603930890 T^{7} + 8040333650992 T^{8} + 268603930890 p^{2} T^{9} + 7979134318 p^{4} T^{10} + 221987952 p^{6} T^{11} + 6239425 p^{8} T^{12} + 128025 p^{10} T^{13} + 3520 p^{12} T^{14} + 45 p^{14} T^{15} + p^{16} T^{16} \) | |
| 37 | \( ( 1 - 10 T + 1720 T^{2} + 76250 T^{3} + 347470 T^{4} + 76250 p^{2} T^{5} + 1720 p^{4} T^{6} - 10 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 41 | \( 1 + 54 T - 1108 T^{2} + 54648 T^{3} + 6254527 T^{4} - 135074304 T^{5} - 346116544 T^{6} + 221393044530 T^{7} - 6391866656792 T^{8} + 221393044530 p^{2} T^{9} - 346116544 p^{4} T^{10} - 135074304 p^{6} T^{11} + 6254527 p^{8} T^{12} + 54648 p^{10} T^{13} - 1108 p^{12} T^{14} + 54 p^{14} T^{15} + p^{16} T^{16} \) | |
| 43 | \( 1 + 4210 T^{2} + 9163465 T^{4} + 282271716 T^{5} + 12018514642 T^{6} + 1023475278060 T^{7} + 9810745543012 T^{8} + 1023475278060 p^{2} T^{9} + 12018514642 p^{4} T^{10} + 282271716 p^{6} T^{11} + 9163465 p^{8} T^{12} + 4210 p^{12} T^{14} + p^{16} T^{16} \) | |
| 47 | \( 1 - 207 T + 24406 T^{2} - 2095461 T^{3} + 142092151 T^{4} - 7940274480 T^{5} + 383592144688 T^{6} - 17253891231516 T^{7} + 786328390118740 T^{8} - 17253891231516 p^{2} T^{9} + 383592144688 p^{4} T^{10} - 7940274480 p^{6} T^{11} + 142092151 p^{8} T^{12} - 2095461 p^{10} T^{13} + 24406 p^{12} T^{14} - 207 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( ( 1 + 126 T + 12208 T^{2} + 787698 T^{3} + 46295070 T^{4} + 787698 p^{2} T^{5} + 12208 p^{4} T^{6} + 126 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 59 | \( 1 - 306 T + 52768 T^{2} - 6596136 T^{3} + 655249567 T^{4} - 54840345300 T^{5} + 4031461665844 T^{6} - 267864455425266 T^{7} + 16404403358884120 T^{8} - 267864455425266 p^{2} T^{9} + 4031461665844 p^{4} T^{10} - 54840345300 p^{6} T^{11} + 655249567 p^{8} T^{12} - 6596136 p^{10} T^{13} + 52768 p^{12} T^{14} - 306 p^{14} T^{15} + p^{16} T^{16} \) | |
| 61 | \( 1 + 7 T - 8478 T^{2} + 347909 T^{3} + 38508983 T^{4} - 2257992504 T^{5} - 39583245248 T^{6} + 5257966541956 T^{7} - 30952228546836 T^{8} + 5257966541956 p^{2} T^{9} - 39583245248 p^{4} T^{10} - 2257992504 p^{6} T^{11} + 38508983 p^{8} T^{12} + 347909 p^{10} T^{13} - 8478 p^{12} T^{14} + 7 p^{14} T^{15} + p^{16} T^{16} \) | |
| 67 | \( 1 - 12 T + 11866 T^{2} - 141816 T^{3} + 66010801 T^{4} - 721613340 T^{5} + 400272803818 T^{6} - 3347530167216 T^{7} + 2268475959596500 T^{8} - 3347530167216 p^{2} T^{9} + 400272803818 p^{4} T^{10} - 721613340 p^{6} T^{11} + 66010801 p^{8} T^{12} - 141816 p^{10} T^{13} + 11866 p^{12} T^{14} - 12 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( 1 - 14120 T^{2} + 150082588 T^{4} - 1080637518872 T^{6} + 6319440918414790 T^{8} - 1080637518872 p^{4} T^{10} + 150082588 p^{8} T^{12} - 14120 p^{12} T^{14} + p^{16} T^{16} \) | |
| 73 | \( ( 1 - 37 T + 20314 T^{2} - 565891 T^{3} + 160126666 T^{4} - 565891 p^{2} T^{5} + 20314 p^{4} T^{6} - 37 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 79 | \( 1 + 33 T + 13924 T^{2} + 447513 T^{3} + 83250565 T^{4} + 4145960160 T^{5} + 461050019242 T^{6} + 35083235957766 T^{7} + 3234248047096936 T^{8} + 35083235957766 p^{2} T^{9} + 461050019242 p^{4} T^{10} + 4145960160 p^{6} T^{11} + 83250565 p^{8} T^{12} + 447513 p^{10} T^{13} + 13924 p^{12} T^{14} + 33 p^{14} T^{15} + p^{16} T^{16} \) | |
| 83 | \( 1 + 549 T + 157876 T^{2} + 31517541 T^{3} + 4848813277 T^{4} + 610140548160 T^{5} + 65668976887426 T^{6} + 6260783050033902 T^{7} + 542374558465652968 T^{8} + 6260783050033902 p^{2} T^{9} + 65668976887426 p^{4} T^{10} + 610140548160 p^{6} T^{11} + 4848813277 p^{8} T^{12} + 31517541 p^{10} T^{13} + 157876 p^{12} T^{14} + 549 p^{14} T^{15} + p^{16} T^{16} \) | |
| 89 | \( ( 1 - 84 T + 30700 T^{2} - 1886940 T^{3} + 359702982 T^{4} - 1886940 p^{2} T^{5} + 30700 p^{4} T^{6} - 84 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 97 | \( 1 + 10 T - 22548 T^{2} + 828296 T^{3} + 272325791 T^{4} - 14518511616 T^{5} - 1388516520704 T^{6} + 83318070893038 T^{7} + 5416086623373288 T^{8} + 83318070893038 p^{2} T^{9} - 1388516520704 p^{4} T^{10} - 14518511616 p^{6} T^{11} + 272325791 p^{8} T^{12} + 828296 p^{10} T^{13} - 22548 p^{12} T^{14} + 10 p^{14} T^{15} + p^{16} T^{16} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.70521426244560024975736686666, −3.48364230162801099136714829466, −3.40597063652790024485229911762, −3.37327481660068235427967850448, −3.25642065701719071032944479220, −3.05363534258135660595953647276, −2.91916727644425601528503164308, −2.85773969978328044466606542317, −2.60583208169307327800904039898, −2.49232247129193097522069327284, −2.43039667812392866373264809907, −2.33805248287173968495269700077, −2.19058546456922655901964040555, −1.97485415215748337525122675331, −1.92353586779068660647735750622, −1.68921806736655414495636179684, −1.68263941884559906278501549907, −1.49665848722082691681094917771, −1.15620954533361650884309499831, −0.989742789447545059699184917652, −0.819305864787609040396363430685, −0.59942045478884677848805183935, −0.58005235056938181333643948466, −0.25147011660944960238556165452, −0.12643044433167658553083552856, 0.12643044433167658553083552856, 0.25147011660944960238556165452, 0.58005235056938181333643948466, 0.59942045478884677848805183935, 0.819305864787609040396363430685, 0.989742789447545059699184917652, 1.15620954533361650884309499831, 1.49665848722082691681094917771, 1.68263941884559906278501549907, 1.68921806736655414495636179684, 1.92353586779068660647735750622, 1.97485415215748337525122675331, 2.19058546456922655901964040555, 2.33805248287173968495269700077, 2.43039667812392866373264809907, 2.49232247129193097522069327284, 2.60583208169307327800904039898, 2.85773969978328044466606542317, 2.91916727644425601528503164308, 3.05363534258135660595953647276, 3.25642065701719071032944479220, 3.37327481660068235427967850448, 3.40597063652790024485229911762, 3.48364230162801099136714829466, 3.70521426244560024975736686666
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.