Dirichlet series
| L(s) = 1 | − 7·4-s + 4·5-s − 12·11-s + 28·16-s + 4·19-s − 28·20-s + 13·25-s + 16·29-s + 12·31-s − 56·41-s + 84·44-s + 58·49-s − 48·55-s + 28·59-s + 12·61-s − 84·64-s − 48·71-s − 28·76-s − 20·79-s + 112·80-s + 36·89-s + 16·95-s − 91·100-s − 92·101-s + 12·109-s − 112·116-s + 6·121-s + ⋯ |
| L(s) = 1 | − 7/2·4-s + 1.78·5-s − 3.61·11-s + 7·16-s + 0.917·19-s − 6.26·20-s + 13/5·25-s + 2.97·29-s + 2.15·31-s − 8.74·41-s + 12.6·44-s + 58/7·49-s − 6.47·55-s + 3.64·59-s + 1.53·61-s − 10.5·64-s − 5.69·71-s − 3.21·76-s − 2.25·79-s + 12.5·80-s + 3.81·89-s + 1.64·95-s − 9.09·100-s − 9.15·101-s + 1.14·109-s − 10.3·116-s + 6/11·121-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{28} \cdot 5^{14} \cdot 37^{14}\right)^{s/2} \, \Gamma_{\C}(s)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{14} \cdot 3^{28} \cdot 5^{14} \cdot 37^{14}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{14} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(28\) |
| Conductor: | \(2^{14} \cdot 3^{28} \cdot 5^{14} \cdot 37^{14}\) |
| Sign: | $1$ |
| Analytic conductor: | \(8.83280\times 10^{19}\) |
| Root analytic conductor: | \(5.15656\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((28,\ 2^{14} \cdot 3^{28} \cdot 5^{14} \cdot 37^{14} ,\ ( \ : [1/2]^{14} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.3819812825\) |
| \(L(\frac12)\) | \(\approx\) | \(0.3819812825\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T^{2} )^{7} \) |
| 3 | \( 1 \) | |
| 5 | \( 1 - 4 T + 3 T^{2} + 16 T^{3} - 31 T^{4} - 28 T^{5} + 99 T^{6} - 96 T^{7} + 99 p T^{8} - 28 p^{2} T^{9} - 31 p^{3} T^{10} + 16 p^{4} T^{11} + 3 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( ( 1 + T^{2} )^{7} \) | |
| good | 7 | \( 1 - 58 T^{2} + 1553 T^{4} - 25124 T^{6} + 270376 T^{8} - 2036260 T^{10} + 1681482 p T^{12} - 70266524 T^{14} + 1681482 p^{3} T^{16} - 2036260 p^{4} T^{18} + 270376 p^{6} T^{20} - 25124 p^{8} T^{22} + 1553 p^{10} T^{24} - 58 p^{12} T^{26} + p^{14} T^{28} \) |
| 11 | \( ( 1 + 6 T + 51 T^{2} + 244 T^{3} + 1280 T^{4} + 5352 T^{5} + 20642 T^{6} + 72520 T^{7} + 20642 p T^{8} + 5352 p^{2} T^{9} + 1280 p^{3} T^{10} + 244 p^{4} T^{11} + 51 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 13 | \( 1 - 110 T^{2} + 6105 T^{4} - 225460 T^{6} + 6175400 T^{8} - 132650100 T^{10} + 2305610950 T^{12} - 32963077588 T^{14} + 2305610950 p^{2} T^{16} - 132650100 p^{4} T^{18} + 6175400 p^{6} T^{20} - 225460 p^{8} T^{22} + 6105 p^{10} T^{24} - 110 p^{12} T^{26} + p^{14} T^{28} \) | |
| 17 | \( 1 - 110 T^{2} + 5849 T^{4} - 205996 T^{6} + 332360 p T^{8} - 133119196 T^{10} + 2780349646 T^{12} - 50815922292 T^{14} + 2780349646 p^{2} T^{16} - 133119196 p^{4} T^{18} + 332360 p^{7} T^{20} - 205996 p^{8} T^{22} + 5849 p^{10} T^{24} - 110 p^{12} T^{26} + p^{14} T^{28} \) | |
| 19 | \( ( 1 - 2 T + 99 T^{2} - 176 T^{3} + 4656 T^{4} - 7156 T^{5} + 134398 T^{6} - 172352 T^{7} + 134398 p T^{8} - 7156 p^{2} T^{9} + 4656 p^{3} T^{10} - 176 p^{4} T^{11} + 99 p^{5} T^{12} - 2 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 23 | \( 1 - 174 T^{2} + 13873 T^{4} - 654876 T^{6} + 19430344 T^{8} - 338326940 T^{10} + 2114815046 T^{12} + 20496877484 T^{14} + 2114815046 p^{2} T^{16} - 338326940 p^{4} T^{18} + 19430344 p^{6} T^{20} - 654876 p^{8} T^{22} + 13873 p^{10} T^{24} - 174 p^{12} T^{26} + p^{14} T^{28} \) | |
| 29 | \( ( 1 - 8 T + 107 T^{2} - 376 T^{3} + 4097 T^{4} - 10120 T^{5} + 153547 T^{6} - 382576 T^{7} + 153547 p T^{8} - 10120 p^{2} T^{9} + 4097 p^{3} T^{10} - 376 p^{4} T^{11} + 107 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 31 | \( ( 1 - 6 T + 141 T^{2} - 516 T^{3} + 271 p T^{4} - 19770 T^{5} + 328709 T^{6} - 599672 T^{7} + 328709 p T^{8} - 19770 p^{2} T^{9} + 271 p^{4} T^{10} - 516 p^{4} T^{11} + 141 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 41 | \( ( 1 + 28 T + 551 T^{2} + 7512 T^{3} + 84737 T^{4} + 777076 T^{5} + 6213143 T^{6} + 42293872 T^{7} + 6213143 p T^{8} + 777076 p^{2} T^{9} + 84737 p^{3} T^{10} + 7512 p^{4} T^{11} + 551 p^{5} T^{12} + 28 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 43 | \( 1 - 286 T^{2} + 43683 T^{4} - 4648172 T^{6} + 8865643 p T^{8} - 25335776290 T^{10} + 1401571838579 T^{12} - 65452109861864 T^{14} + 1401571838579 p^{2} T^{16} - 25335776290 p^{4} T^{18} + 8865643 p^{7} T^{20} - 4648172 p^{8} T^{22} + 43683 p^{10} T^{24} - 286 p^{12} T^{26} + p^{14} T^{28} \) | |
| 47 | \( 1 - 418 T^{2} + 86467 T^{4} - 11785940 T^{6} + 1185321761 T^{8} - 93125459614 T^{10} + 5895522209243 T^{12} - 305518144245720 T^{14} + 5895522209243 p^{2} T^{16} - 93125459614 p^{4} T^{18} + 1185321761 p^{6} T^{20} - 11785940 p^{8} T^{22} + 86467 p^{10} T^{24} - 418 p^{12} T^{26} + p^{14} T^{28} \) | |
| 53 | \( 1 - 114 T^{2} + 16409 T^{4} - 1494460 T^{6} + 128764776 T^{8} - 8969346908 T^{10} + 588964237302 T^{12} - 31812064268588 T^{14} + 588964237302 p^{2} T^{16} - 8969346908 p^{4} T^{18} + 128764776 p^{6} T^{20} - 1494460 p^{8} T^{22} + 16409 p^{10} T^{24} - 114 p^{12} T^{26} + p^{14} T^{28} \) | |
| 59 | \( ( 1 - 14 T + 297 T^{2} - 2364 T^{3} + 27349 T^{4} - 110986 T^{5} + 1129349 T^{6} - 2385816 T^{7} + 1129349 p T^{8} - 110986 p^{2} T^{9} + 27349 p^{3} T^{10} - 2364 p^{4} T^{11} + 297 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 61 | \( ( 1 - 6 T + 319 T^{2} - 2136 T^{3} + 46273 T^{4} - 320314 T^{5} + 4119719 T^{6} - 25905008 T^{7} + 4119719 p T^{8} - 320314 p^{2} T^{9} + 46273 p^{3} T^{10} - 2136 p^{4} T^{11} + 319 p^{5} T^{12} - 6 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 67 | \( 1 - 410 T^{2} + 85939 T^{4} - 11747524 T^{6} + 1155635129 T^{8} - 87490773222 T^{10} + 5615506636035 T^{12} - 358461148176568 T^{14} + 5615506636035 p^{2} T^{16} - 87490773222 p^{4} T^{18} + 1155635129 p^{6} T^{20} - 11747524 p^{8} T^{22} + 85939 p^{10} T^{24} - 410 p^{12} T^{26} + p^{14} T^{28} \) | |
| 71 | \( ( 1 + 24 T + 529 T^{2} + 7872 T^{3} + 104969 T^{4} + 16376 p T^{5} + 11674521 T^{6} + 103077952 T^{7} + 11674521 p T^{8} + 16376 p^{3} T^{9} + 104969 p^{3} T^{10} + 7872 p^{4} T^{11} + 529 p^{5} T^{12} + 24 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 73 | \( 1 - 626 T^{2} + 187513 T^{4} - 36353380 T^{6} + 5217206216 T^{8} - 597368497236 T^{10} + 56660191116222 T^{12} - 4510134198046348 T^{14} + 56660191116222 p^{2} T^{16} - 597368497236 p^{4} T^{18} + 5217206216 p^{6} T^{20} - 36353380 p^{8} T^{22} + 187513 p^{10} T^{24} - 626 p^{12} T^{26} + p^{14} T^{28} \) | |
| 79 | \( ( 1 + 10 T + 269 T^{2} + 2196 T^{3} + 27285 T^{4} + 136782 T^{5} + 1502121 T^{6} + 4068424 T^{7} + 1502121 p T^{8} + 136782 p^{2} T^{9} + 27285 p^{3} T^{10} + 2196 p^{4} T^{11} + 269 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 83 | \( 1 - 886 T^{2} + 374561 T^{4} - 100594236 T^{6} + 19281014664 T^{8} - 2807443716300 T^{10} + 322143697048734 T^{12} - 29721849571873156 T^{14} + 322143697048734 p^{2} T^{16} - 2807443716300 p^{4} T^{18} + 19281014664 p^{6} T^{20} - 100594236 p^{8} T^{22} + 374561 p^{10} T^{24} - 886 p^{12} T^{26} + p^{14} T^{28} \) | |
| 89 | \( ( 1 - 18 T + 509 T^{2} - 6476 T^{3} + 106740 T^{4} - 1063844 T^{5} + 13466206 T^{6} - 112338076 T^{7} + 13466206 p T^{8} - 1063844 p^{2} T^{9} + 106740 p^{3} T^{10} - 6476 p^{4} T^{11} + 509 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} )^{2} \) | |
| 97 | \( 1 - 574 T^{2} + 191299 T^{4} - 44091980 T^{6} + 7879071873 T^{8} - 1139872654210 T^{10} + 139014907213243 T^{12} - 14498191144131304 T^{14} + 139014907213243 p^{2} T^{16} - 1139872654210 p^{4} T^{18} + 7879071873 p^{6} T^{20} - 44091980 p^{8} T^{22} + 191299 p^{10} T^{24} - 574 p^{12} T^{26} + p^{14} T^{28} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{28} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.31126235725857293059668935015, −2.18006580516240574958033037320, −2.14093040305771325450125747467, −2.01794611843310890496081741035, −1.94408616813990193234260881755, −1.92430627899067913063347966257, −1.91655553785474102552929606667, −1.85182900104443684445986690308, −1.85029251384377574588498726481, −1.56927542164419525541655200765, −1.38719752952900180318332140191, −1.31692196960714872109656469334, −1.28891081015004141903220381230, −1.26740619042721097899710099743, −1.09067296904010879977508896757, −1.03407233113350389293143343482, −1.01390568814860800563555854520, −0.834237899254373491560863718884, −0.75714553142405828056708993156, −0.73069690479785477573737672906, −0.67634109286920921058013063518, −0.32389845506313966987776688615, −0.25699819513077687933271079826, −0.15912950019569511229388536604, −0.079072587717713775034357627849, 0.079072587717713775034357627849, 0.15912950019569511229388536604, 0.25699819513077687933271079826, 0.32389845506313966987776688615, 0.67634109286920921058013063518, 0.73069690479785477573737672906, 0.75714553142405828056708993156, 0.834237899254373491560863718884, 1.01390568814860800563555854520, 1.03407233113350389293143343482, 1.09067296904010879977508896757, 1.26740619042721097899710099743, 1.28891081015004141903220381230, 1.31692196960714872109656469334, 1.38719752952900180318332140191, 1.56927542164419525541655200765, 1.85029251384377574588498726481, 1.85182900104443684445986690308, 1.91655553785474102552929606667, 1.92430627899067913063347966257, 1.94408616813990193234260881755, 2.01794611843310890496081741035, 2.14093040305771325450125747467, 2.18006580516240574958033037320, 2.31126235725857293059668935015
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.