Dirichlet series
| L(s) = 1 | − 5·5-s − 5·11-s + 5·17-s − 8·19-s + 20·23-s + 10·25-s + 8·29-s − 12·31-s − 10·37-s − 13·41-s − 45·47-s + 35·49-s − 30·53-s + 25·55-s − 9·59-s + 16·61-s − 5·67-s + 71-s − 60·73-s − 24·79-s + 10·83-s − 25·85-s + 37·89-s + 40·95-s + 25·97-s + 72·101-s + 10·103-s + ⋯ |
| L(s) = 1 | − 2.23·5-s − 1.50·11-s + 1.21·17-s − 1.83·19-s + 4.17·23-s + 2·25-s + 1.48·29-s − 2.15·31-s − 1.64·37-s − 2.03·41-s − 6.56·47-s + 5·49-s − 4.12·53-s + 3.37·55-s − 1.17·59-s + 2.04·61-s − 0.610·67-s + 0.118·71-s − 7.02·73-s − 2.70·79-s + 1.09·83-s − 2.71·85-s + 3.92·89-s + 4.10·95-s + 2.53·97-s + 7.16·101-s + 0.985·103-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{16} \cdot 5^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{16} \cdot 3^{16} \cdot 5^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.11470\times 10^{6}\) |
| Root analytic conductor: | \(2.68077\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{16} \cdot 3^{16} \cdot 5^{16} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.8497393499\) |
| \(L(\frac12)\) | \(\approx\) | \(0.8497393499\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 \) | |
| 5 | \( 1 + p T + 3 p T^{2} + 7 p T^{3} + 16 p T^{4} + 7 p^{2} T^{5} + 3 p^{3} T^{6} + p^{4} T^{7} + p^{4} T^{8} \) | |
| good | 7 | \( 1 - 5 p T^{2} + 551 T^{4} - 775 p T^{6} + 41176 T^{8} - 775 p^{3} T^{10} + 551 p^{4} T^{12} - 5 p^{7} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 + 5 T + 8 T^{2} + 20 T^{3} + 188 T^{4} + 655 T^{5} + 681 T^{6} + 3600 T^{7} + 23500 T^{8} + 3600 p T^{9} + 681 p^{2} T^{10} + 655 p^{3} T^{11} + 188 p^{4} T^{12} + 20 p^{5} T^{13} + 8 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 7 T^{2} + 50 T^{3} + 105 T^{4} + 350 T^{5} + 2317 T^{6} + 13880 T^{7} + 12484 T^{8} + 13880 p T^{9} + 2317 p^{2} T^{10} + 350 p^{3} T^{11} + 105 p^{4} T^{12} + 50 p^{5} T^{13} + 7 p^{6} T^{14} + p^{8} T^{16} \) | |
| 17 | \( 1 - 5 T + 18 T^{2} - 60 T^{3} + 20 T^{4} + 315 T^{5} - 5207 T^{6} + 17930 T^{7} - 24376 T^{8} + 17930 p T^{9} - 5207 p^{2} T^{10} + 315 p^{3} T^{11} + 20 p^{4} T^{12} - 60 p^{5} T^{13} + 18 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 8 T + 55 T^{2} + 330 T^{3} + 1905 T^{4} + 10214 T^{5} + 50917 T^{6} + 225500 T^{7} + 1025380 T^{8} + 225500 p T^{9} + 50917 p^{2} T^{10} + 10214 p^{3} T^{11} + 1905 p^{4} T^{12} + 330 p^{5} T^{13} + 55 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 20 T + 180 T^{2} - 695 T^{3} - 1459 T^{4} + 30080 T^{5} - 120345 T^{6} - 100285 T^{7} + 2412686 T^{8} - 100285 p T^{9} - 120345 p^{2} T^{10} + 30080 p^{3} T^{11} - 1459 p^{4} T^{12} - 695 p^{5} T^{13} + 180 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 8 T + 5 T^{2} + 200 T^{3} - 845 T^{4} - 8104 T^{5} + 60447 T^{6} + 21800 T^{7} - 1138800 T^{8} + 21800 p T^{9} + 60447 p^{2} T^{10} - 8104 p^{3} T^{11} - 845 p^{4} T^{12} + 200 p^{5} T^{13} + 5 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 12 T - 4 T^{2} - 17 p T^{3} - 929 T^{4} + 16076 T^{5} + 97629 T^{6} - 73041 T^{7} - 2777218 T^{8} - 73041 p T^{9} + 97629 p^{2} T^{10} + 16076 p^{3} T^{11} - 929 p^{4} T^{12} - 17 p^{6} T^{13} - 4 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 10 T + 140 T^{2} + 1685 T^{3} + 12811 T^{4} + 3230 p T^{5} + 861615 T^{6} + 5562535 T^{7} + 38923586 T^{8} + 5562535 p T^{9} + 861615 p^{2} T^{10} + 3230 p^{4} T^{11} + 12811 p^{4} T^{12} + 1685 p^{5} T^{13} + 140 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 13 T + 36 T^{2} - 388 T^{3} - 4124 T^{4} - 25421 T^{5} - 46041 T^{6} + 1100136 T^{7} + 12326712 T^{8} + 1100136 p T^{9} - 46041 p^{2} T^{10} - 25421 p^{3} T^{11} - 4124 p^{4} T^{12} - 388 p^{5} T^{13} + 36 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 219 T^{2} + 21607 T^{4} - 1334737 T^{6} + 62871560 T^{8} - 1334737 p^{2} T^{10} + 21607 p^{4} T^{12} - 219 p^{6} T^{14} + p^{8} T^{16} \) | |
| 47 | \( 1 + 45 T + 938 T^{2} + 11970 T^{3} + 107730 T^{4} + 813105 T^{5} + 6411283 T^{6} + 1150740 p T^{7} + 409062224 T^{8} + 1150740 p^{2} T^{9} + 6411283 p^{2} T^{10} + 813105 p^{3} T^{11} + 107730 p^{4} T^{12} + 11970 p^{5} T^{13} + 938 p^{6} T^{14} + 45 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 30 T + 600 T^{2} + 8895 T^{3} + 108851 T^{4} + 1137030 T^{5} + 10519575 T^{6} + 87403425 T^{7} + 665807146 T^{8} + 87403425 p T^{9} + 10519575 p^{2} T^{10} + 1137030 p^{3} T^{11} + 108851 p^{4} T^{12} + 8895 p^{5} T^{13} + 600 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 9 T - 16 T^{2} + 534 T^{3} + 11816 T^{4} + 58317 T^{5} + 169591 T^{6} + 4043232 T^{7} + 51996952 T^{8} + 4043232 p T^{9} + 169591 p^{2} T^{10} + 58317 p^{3} T^{11} + 11816 p^{4} T^{12} + 534 p^{5} T^{13} - 16 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 16 T + 1485 T^{3} - 8915 T^{4} + 512 p T^{5} - 4137 p T^{6} - 985 p^{2} T^{7} + 19230 p^{2} T^{8} - 985 p^{3} T^{9} - 4137 p^{3} T^{10} + 512 p^{4} T^{11} - 8915 p^{4} T^{12} + 1485 p^{5} T^{13} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 5 T + 180 T^{2} + 990 T^{3} + 12196 T^{4} + 74305 T^{5} + 246935 T^{6} + 2417280 T^{7} - 7902424 T^{8} + 2417280 p T^{9} + 246935 p^{2} T^{10} + 74305 p^{3} T^{11} + 12196 p^{4} T^{12} + 990 p^{5} T^{13} + 180 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - T + 170 T^{2} - 410 T^{3} + 25030 T^{4} - 53273 T^{5} + 2381343 T^{6} - 5547800 T^{7} + 194260320 T^{8} - 5547800 p T^{9} + 2381343 p^{2} T^{10} - 53273 p^{3} T^{11} + 25030 p^{4} T^{12} - 410 p^{5} T^{13} + 170 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 60 T + 1772 T^{2} + 33525 T^{3} + 445815 T^{4} + 4298160 T^{5} + 30003487 T^{6} + 155927835 T^{7} + 910902194 T^{8} + 155927835 p T^{9} + 30003487 p^{2} T^{10} + 4298160 p^{3} T^{11} + 445815 p^{4} T^{12} + 33525 p^{5} T^{13} + 1772 p^{6} T^{14} + 60 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 24 T + 99 T^{2} - 1326 T^{3} - 979 T^{4} + 289302 T^{5} + 3270321 T^{6} + 4493532 T^{7} - 130612868 T^{8} + 4493532 p T^{9} + 3270321 p^{2} T^{10} + 289302 p^{3} T^{11} - 979 p^{4} T^{12} - 1326 p^{5} T^{13} + 99 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 10 T + 180 T^{2} + 815 T^{3} - 7979 T^{4} + 308990 T^{5} + 290165 T^{6} - 231275 T^{7} + 274435626 T^{8} - 231275 p T^{9} + 290165 p^{2} T^{10} + 308990 p^{3} T^{11} - 7979 p^{4} T^{12} + 815 p^{5} T^{13} + 180 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 37 T + 590 T^{2} - 5480 T^{3} + 34820 T^{4} - 246401 T^{5} + 3372057 T^{6} - 45787390 T^{7} + 479697120 T^{8} - 45787390 p T^{9} + 3372057 p^{2} T^{10} - 246401 p^{3} T^{11} + 34820 p^{4} T^{12} - 5480 p^{5} T^{13} + 590 p^{6} T^{14} - 37 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 25 T + 240 T^{2} + 1020 T^{3} - 53264 T^{4} + 582025 T^{5} - 1401985 T^{6} - 44565840 T^{7} + 691186376 T^{8} - 44565840 p T^{9} - 1401985 p^{2} T^{10} + 582025 p^{3} T^{11} - 53264 p^{4} T^{12} + 1020 p^{5} T^{13} + 240 p^{6} T^{14} - 25 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.57250492452564824869299681762, −4.17757633271860774296016014944, −4.01934878940461513166605763475, −3.99600761318506640688618202147, −3.80200268274400496705298845468, −3.48146368004818221533435437582, −3.43798565725554687399510114663, −3.33092062346532941821351229892, −3.31919096372379149557394486753, −3.18580436435768678732032226951, −3.14699450726301553583195946477, −2.98600924972611855226312102326, −2.85836557271675787625242390046, −2.59678543659090271685926430074, −2.27256342235105723058436067360, −2.22436947973634189343680014056, −2.05398561342899122199429921520, −1.80800405097121454068947595428, −1.68516210267397254029687507092, −1.37284864077404092573568313034, −1.27385219791379005719223903726, −1.17977293481925354414564570811, −0.48875142972675425680396516956, −0.46352121631887218033193957278, −0.23354960575923066823149936323, 0.23354960575923066823149936323, 0.46352121631887218033193957278, 0.48875142972675425680396516956, 1.17977293481925354414564570811, 1.27385219791379005719223903726, 1.37284864077404092573568313034, 1.68516210267397254029687507092, 1.80800405097121454068947595428, 2.05398561342899122199429921520, 2.22436947973634189343680014056, 2.27256342235105723058436067360, 2.59678543659090271685926430074, 2.85836557271675787625242390046, 2.98600924972611855226312102326, 3.14699450726301553583195946477, 3.18580436435768678732032226951, 3.31919096372379149557394486753, 3.33092062346532941821351229892, 3.43798565725554687399510114663, 3.48146368004818221533435437582, 3.80200268274400496705298845468, 3.99600761318506640688618202147, 4.01934878940461513166605763475, 4.17757633271860774296016014944, 4.57250492452564824869299681762
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.