Dirichlet series
| L(s) = 1 | + 4·3-s − 2·5-s + 6·9-s − 8·15-s + 4·17-s + 6·23-s + 6·25-s − 12·29-s − 28·31-s − 12·41-s − 18·43-s − 12·45-s + 12·47-s + 16·49-s + 16·51-s − 6·53-s + 10·59-s − 4·61-s + 6·67-s + 24·69-s − 8·71-s − 8·73-s + 24·75-s + 14·79-s − 15·81-s − 8·85-s − 48·87-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 0.894·5-s + 2·9-s − 2.06·15-s + 0.970·17-s + 1.25·23-s + 6/5·25-s − 2.22·29-s − 5.02·31-s − 1.87·41-s − 2.74·43-s − 1.78·45-s + 1.75·47-s + 16/7·49-s + 2.24·51-s − 0.824·53-s + 1.30·59-s − 0.512·61-s + 0.733·67-s + 2.88·69-s − 0.949·71-s − 0.936·73-s + 2.77·75-s + 1.57·79-s − 5/3·81-s − 0.867·85-s − 5.14·87-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 3^{8} \cdot 19^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(7.90998\times 10^{6}\) |
| Root analytic conductor: | \(2.69858\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 3^{8} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(3.146237153\) |
| \(L(\frac12)\) | \(\approx\) | \(3.146237153\) |
| \(L(\frac{3}{2})\) | 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 | \( ( 1 - T + T^{2} )^{4} \) | |
| 19 | \( 1 - 4 T^{2} - 72 T^{3} + 258 T^{4} - 72 p T^{5} - 4 p^{2} T^{6} + p^{4} T^{8} \) | |
| good | 5 | \( 1 + 2 T - 2 T^{2} + 24 T^{3} + 56 T^{4} - 34 T^{5} + 444 T^{6} + 958 T^{7} - 641 T^{8} + 958 p T^{9} + 444 p^{2} T^{10} - 34 p^{3} T^{11} + 56 p^{4} T^{12} + 24 p^{5} T^{13} - 2 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - 16 T^{2} + 110 T^{4} - 456 T^{6} + 1787 T^{8} - 456 p^{2} T^{10} + 110 p^{4} T^{12} - 16 p^{6} T^{14} + p^{8} T^{16} \) | |
| 11 | \( 1 - 48 T^{2} + 1268 T^{4} - 2040 p T^{6} + 288234 T^{8} - 2040 p^{3} T^{10} + 1268 p^{4} T^{12} - 48 p^{6} T^{14} + p^{8} T^{16} \) | |
| 13 | \( 1 + 2 p T^{2} + 353 T^{4} + 72 p T^{5} + 1338 T^{6} + 23256 T^{7} - 16012 T^{8} + 23256 p T^{9} + 1338 p^{2} T^{10} + 72 p^{4} T^{11} + 353 p^{4} T^{12} + 2 p^{7} T^{14} + p^{8} T^{16} \) | |
| 17 | \( 1 - 4 T - 8 T^{2} - 216 T^{3} + 938 T^{4} + 1484 T^{5} + 1440 p T^{6} - 103676 T^{7} - 139085 T^{8} - 103676 p T^{9} + 1440 p^{3} T^{10} + 1484 p^{3} T^{11} + 938 p^{4} T^{12} - 216 p^{5} T^{13} - 8 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 6 T + 66 T^{2} - 324 T^{3} + 2252 T^{4} - 13578 T^{5} + 132 p^{2} T^{6} - 430122 T^{7} + 1727991 T^{8} - 430122 p T^{9} + 132 p^{4} T^{10} - 13578 p^{3} T^{11} + 2252 p^{4} T^{12} - 324 p^{5} T^{13} + 66 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 12 T + 108 T^{2} + 720 T^{3} + 3386 T^{4} + 5580 T^{5} - 54960 T^{6} - 730068 T^{7} - 4890669 T^{8} - 730068 p T^{9} - 54960 p^{2} T^{10} + 5580 p^{3} T^{11} + 3386 p^{4} T^{12} + 720 p^{5} T^{13} + 108 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( ( 1 + 14 T + 134 T^{2} + 1020 T^{3} + 6341 T^{4} + 1020 p T^{5} + 134 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 37 | \( 1 - 4 p T^{2} + 11522 T^{4} - 649824 T^{6} + 27865931 T^{8} - 649824 p^{2} T^{10} + 11522 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \) | |
| 41 | \( 1 + 12 T + 152 T^{2} + 1248 T^{3} + 9322 T^{4} + 52164 T^{5} + 242912 T^{6} + 1054428 T^{7} + 4843411 T^{8} + 1054428 p T^{9} + 242912 p^{2} T^{10} + 52164 p^{3} T^{11} + 9322 p^{4} T^{12} + 1248 p^{5} T^{13} + 152 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 18 T + 214 T^{2} + 1908 T^{3} + 12631 T^{4} + 86184 T^{5} + 13978 p T^{6} + 4361166 T^{7} + 31873036 T^{8} + 4361166 p T^{9} + 13978 p^{3} T^{10} + 86184 p^{3} T^{11} + 12631 p^{4} T^{12} + 1908 p^{5} T^{13} + 214 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 12 T + 168 T^{2} - 1440 T^{3} + 12650 T^{4} - 93732 T^{5} + 604320 T^{6} - 4414788 T^{7} + 26589651 T^{8} - 4414788 p T^{9} + 604320 p^{2} T^{10} - 93732 p^{3} T^{11} + 12650 p^{4} T^{12} - 1440 p^{5} T^{13} + 168 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 6 T + 210 T^{2} + 1188 T^{3} + 25640 T^{4} + 131370 T^{5} + 2109660 T^{6} + 9700302 T^{7} + 129335247 T^{8} + 9700302 p T^{9} + 2109660 p^{2} T^{10} + 131370 p^{3} T^{11} + 25640 p^{4} T^{12} + 1188 p^{5} T^{13} + 210 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 10 T - 38 T^{2} + 936 T^{3} - 5620 T^{4} + 33554 T^{5} - 187092 T^{6} - 3151970 T^{7} + 54740455 T^{8} - 3151970 p T^{9} - 187092 p^{2} T^{10} + 33554 p^{3} T^{11} - 5620 p^{4} T^{12} + 936 p^{5} T^{13} - 38 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 4 T - 90 T^{2} - 40 T^{3} + 3557 T^{4} - 15264 T^{5} + 70462 T^{6} + 1051636 T^{7} - 6098868 T^{8} + 1051636 p T^{9} + 70462 p^{2} T^{10} - 15264 p^{3} T^{11} + 3557 p^{4} T^{12} - 40 p^{5} T^{13} - 90 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 6 T - 26 T^{2} - 732 T^{3} + 6427 T^{4} + 5184 T^{5} + 754582 T^{6} - 5240466 T^{7} - 15663692 T^{8} - 5240466 p T^{9} + 754582 p^{2} T^{10} + 5184 p^{3} T^{11} + 6427 p^{4} T^{12} - 732 p^{5} T^{13} - 26 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 8 T - 152 T^{2} - 1440 T^{3} + 12602 T^{4} + 112496 T^{5} - 831456 T^{6} - 3141224 T^{7} + 64254979 T^{8} - 3141224 p T^{9} - 831456 p^{2} T^{10} + 112496 p^{3} T^{11} + 12602 p^{4} T^{12} - 1440 p^{5} T^{13} - 152 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 8 T - 190 T^{2} - 704 T^{3} + 27773 T^{4} + 34552 T^{5} - 2773302 T^{6} - 1497216 T^{7} + 206564620 T^{8} - 1497216 p T^{9} - 2773302 p^{2} T^{10} + 34552 p^{3} T^{11} + 27773 p^{4} T^{12} - 704 p^{5} T^{13} - 190 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 14 T - 66 T^{2} + 644 T^{3} + 11855 T^{4} + 9744 T^{5} - 1436426 T^{6} + 2094862 T^{7} + 66521484 T^{8} + 2094862 p T^{9} - 1436426 p^{2} T^{10} + 9744 p^{3} T^{11} + 11855 p^{4} T^{12} + 644 p^{5} T^{13} - 66 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 160 T^{2} + 14188 T^{4} - 1709728 T^{6} + 189871366 T^{8} - 1709728 p^{2} T^{10} + 14188 p^{4} T^{12} - 160 p^{6} T^{14} + p^{8} T^{16} \) | |
| 89 | \( 1 - 54 T + 1530 T^{2} - 30132 T^{3} + 461336 T^{4} - 5878446 T^{5} + 65654868 T^{6} - 671327862 T^{7} + 6482561007 T^{8} - 671327862 p T^{9} + 65654868 p^{2} T^{10} - 5878446 p^{3} T^{11} + 461336 p^{4} T^{12} - 30132 p^{5} T^{13} + 1530 p^{6} T^{14} - 54 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 60 T + 2012 T^{2} - 48720 T^{3} + 939494 T^{4} - 15092220 T^{5} + 207598272 T^{6} - 2484380220 T^{7} + 26090169059 T^{8} - 2484380220 p T^{9} + 207598272 p^{2} T^{10} - 15092220 p^{3} T^{11} + 939494 p^{4} T^{12} - 48720 p^{5} T^{13} + 2012 p^{6} T^{14} - 60 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.23542161926562645513004164028, −3.94910938185792597093593817693, −3.89578626331130656264703065244, −3.77592921552178674439235497726, −3.75456149066034382752091150134, −3.72776172754865836507252088972, −3.55466411528318690013530996280, −3.38502962399243807081535213154, −3.30332918415263698512932721635, −3.15891123996725230497675386917, −3.02835007609455180784964779861, −2.87039429490317797491541952840, −2.86470306940474261267282452721, −2.55474003888166128266468263951, −2.14690688734284147139537249660, −2.12641758816385493846149897216, −2.03922577471347382839157833896, −1.91479199286248973755654268948, −1.88540509896002475724615135974, −1.80815551003577257559923229942, −1.15575241465002928476298032311, −1.13093230717797165813850324578, −0.981934221390499237518095951362, −0.46028971149832108218267522082, −0.20061494322440723420235430499, 0.20061494322440723420235430499, 0.46028971149832108218267522082, 0.981934221390499237518095951362, 1.13093230717797165813850324578, 1.15575241465002928476298032311, 1.80815551003577257559923229942, 1.88540509896002475724615135974, 1.91479199286248973755654268948, 2.03922577471347382839157833896, 2.12641758816385493846149897216, 2.14690688734284147139537249660, 2.55474003888166128266468263951, 2.86470306940474261267282452721, 2.87039429490317797491541952840, 3.02835007609455180784964779861, 3.15891123996725230497675386917, 3.30332918415263698512932721635, 3.38502962399243807081535213154, 3.55466411528318690013530996280, 3.72776172754865836507252088972, 3.75456149066034382752091150134, 3.77592921552178674439235497726, 3.89578626331130656264703065244, 3.94910938185792597093593817693, 4.23542161926562645513004164028
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.