Dirichlet series
| L(s) = 1 | + 3-s − 3·5-s − 4·7-s − 7·11-s + 3·13-s − 3·15-s + 6·17-s + 8·19-s − 4·21-s − 2·23-s + 12·25-s − 4·27-s − 9·29-s − 3·31-s − 7·33-s + 12·35-s + 6·37-s + 3·39-s + 9·41-s − 8·43-s − 3·47-s + 6·49-s + 6·51-s + 12·53-s + 21·55-s + 8·57-s − 10·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.34·5-s − 1.51·7-s − 2.11·11-s + 0.832·13-s − 0.774·15-s + 1.45·17-s + 1.83·19-s − 0.872·21-s − 0.417·23-s + 12/5·25-s − 0.769·27-s − 1.67·29-s − 0.538·31-s − 1.21·33-s + 2.02·35-s + 0.986·37-s + 0.480·39-s + 1.40·41-s − 1.21·43-s − 0.437·47-s + 6/7·49-s + 0.840·51-s + 1.64·53-s + 2.83·55-s + 1.05·57-s − 1.30·59-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{16} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 3^{16} \cdot 7^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.76157\times 10^{7}\) |
| Root analytic conductor: | \(2.83706\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{32} \cdot 3^{16} \cdot 7^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.615460996\) |
| \(L(\frac12)\) | \(\approx\) | \(1.615460996\) |
| \(L(\frac{3}{2})\) | 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 | \( 1 - T + T^{2} + p T^{3} + p T^{4} + p^{2} T^{5} + p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | |
| 7 | \( ( 1 + T + T^{2} )^{4} \) | |
| good | 5 | \( 1 + 3 T - 3 T^{2} - 36 T^{3} - 32 T^{4} + 204 T^{5} + 108 p T^{6} - 597 T^{7} - 3831 T^{8} - 597 p T^{9} + 108 p^{3} T^{10} + 204 p^{3} T^{11} - 32 p^{4} T^{12} - 36 p^{5} T^{13} - 3 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 + 7 T - 2 T^{2} - 81 T^{3} + 203 T^{4} + 1468 T^{5} - 675 T^{6} - 3769 T^{7} + 21730 T^{8} - 3769 p T^{9} - 675 p^{2} T^{10} + 1468 p^{3} T^{11} + 203 p^{4} T^{12} - 81 p^{5} T^{13} - 2 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 3 T - 32 T^{2} + 57 T^{3} + 673 T^{4} - 432 T^{5} - 11810 T^{6} + 2142 T^{7} + 166240 T^{8} + 2142 p T^{9} - 11810 p^{2} T^{10} - 432 p^{3} T^{11} + 673 p^{4} T^{12} + 57 p^{5} T^{13} - 32 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( ( 1 - 3 T + 41 T^{2} - 189 T^{3} + 807 T^{4} - 189 p T^{5} + 41 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 19 | \( ( 1 - 4 T + 49 T^{2} - 193 T^{3} + 1297 T^{4} - 193 p T^{5} + 49 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 23 | \( 1 + 2 T - 59 T^{2} - 192 T^{3} + 1775 T^{4} + 6170 T^{5} - 33567 T^{6} - 71510 T^{7} + 655807 T^{8} - 71510 p T^{9} - 33567 p^{2} T^{10} + 6170 p^{3} T^{11} + 1775 p^{4} T^{12} - 192 p^{5} T^{13} - 59 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 9 T + 13 T^{2} + 72 T^{3} + 904 T^{4} + 2970 T^{5} + 29086 T^{6} + 17613 T^{7} - 1095047 T^{8} + 17613 p T^{9} + 29086 p^{2} T^{10} + 2970 p^{3} T^{11} + 904 p^{4} T^{12} + 72 p^{5} T^{13} + 13 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 3 T - 41 T^{2} - 354 T^{3} - 128 T^{4} + 9180 T^{5} + 42568 T^{6} - 75933 T^{7} - 1254023 T^{8} - 75933 p T^{9} + 42568 p^{2} T^{10} + 9180 p^{3} T^{11} - 128 p^{4} T^{12} - 354 p^{5} T^{13} - 41 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( ( 1 - 3 T + 140 T^{2} - 324 T^{3} + 7620 T^{4} - 324 p T^{5} + 140 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 41 | \( 1 - 9 T - 36 T^{2} + 381 T^{3} + 895 T^{4} + 852 T^{5} - 99027 T^{6} - 57981 T^{7} + 4458864 T^{8} - 57981 p T^{9} - 99027 p^{2} T^{10} + 852 p^{3} T^{11} + 895 p^{4} T^{12} + 381 p^{5} T^{13} - 36 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 8 T - 73 T^{2} - 914 T^{3} + 2642 T^{4} + 51508 T^{5} + 63012 T^{6} - 1050915 T^{7} - 5977292 T^{8} - 1050915 p T^{9} + 63012 p^{2} T^{10} + 51508 p^{3} T^{11} + 2642 p^{4} T^{12} - 914 p^{5} T^{13} - 73 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 3 T - 3 p T^{2} - 486 T^{3} + 11422 T^{4} + 34152 T^{5} - 635364 T^{6} - 765849 T^{7} + 30578781 T^{8} - 765849 p T^{9} - 635364 p^{2} T^{10} + 34152 p^{3} T^{11} + 11422 p^{4} T^{12} - 486 p^{5} T^{13} - 3 p^{7} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( ( 1 - 6 T + 59 T^{2} - 621 T^{3} + 4704 T^{4} - 621 p T^{5} + 59 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 59 | \( 1 + 10 T - 71 T^{2} - 1536 T^{3} - 346 T^{4} + 106150 T^{5} + 583014 T^{6} - 3064183 T^{7} - 49833644 T^{8} - 3064183 p T^{9} + 583014 p^{2} T^{10} + 106150 p^{3} T^{11} - 346 p^{4} T^{12} - 1536 p^{5} T^{13} - 71 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 20 T + 161 T^{2} - 562 T^{3} - 931 T^{4} + 2180 T^{5} - 139869 T^{6} + 5694174 T^{7} - 64208639 T^{8} + 5694174 p T^{9} - 139869 p^{2} T^{10} + 2180 p^{3} T^{11} - 931 p^{4} T^{12} - 562 p^{5} T^{13} + 161 p^{6} T^{14} - 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 11 T - 82 T^{2} - 689 T^{3} + 8039 T^{4} + 18292 T^{5} - 590799 T^{6} - 1428153 T^{7} + 17009218 T^{8} - 1428153 p T^{9} - 590799 p^{2} T^{10} + 18292 p^{3} T^{11} + 8039 p^{4} T^{12} - 689 p^{5} T^{13} - 82 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( ( 1 - 3 T + 276 T^{2} - 630 T^{3} + 29126 T^{4} - 630 p T^{5} + 276 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 73 | \( ( 1 + 24 T + 425 T^{2} + 4923 T^{3} + 48495 T^{4} + 4923 p T^{5} + 425 p^{2} T^{6} + 24 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 79 | \( 1 + 21 T + 133 T^{2} - 252 T^{3} - 9896 T^{4} - 136080 T^{5} - 973784 T^{6} + 2121567 T^{7} + 81625693 T^{8} + 2121567 p T^{9} - 973784 p^{2} T^{10} - 136080 p^{3} T^{11} - 9896 p^{4} T^{12} - 252 p^{5} T^{13} + 133 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 8 T - 259 T^{2} - 1146 T^{3} + 49577 T^{4} + 119878 T^{5} - 6175577 T^{6} - 3335418 T^{7} + 604903483 T^{8} - 3335418 p T^{9} - 6175577 p^{2} T^{10} + 119878 p^{3} T^{11} + 49577 p^{4} T^{12} - 1146 p^{5} T^{13} - 259 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 - 6 T + 263 T^{2} - 1377 T^{3} + 32646 T^{4} - 1377 p T^{5} + 263 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - 16 T - 175 T^{2} + 2086 T^{3} + 49490 T^{4} - 309968 T^{5} - 6453258 T^{6} + 6735477 T^{7} + 823825414 T^{8} + 6735477 p T^{9} - 6453258 p^{2} T^{10} - 309968 p^{3} T^{11} + 49490 p^{4} T^{12} + 2086 p^{5} T^{13} - 175 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.27819902713521196060327028478, −4.16690213423016237680055152272, −3.86848390508924113478945668033, −3.85083996976866560884685640932, −3.59688393453420917764806683674, −3.55892345482151924433836672681, −3.54494205123792899073369566456, −3.21696094288303624487013342666, −3.16723053330200251047595758924, −3.16200537274495010437109373348, −3.04722589507455425534693724351, −2.86875618377608157782837725252, −2.57422703179546698489388327359, −2.52434041632518813850684728423, −2.44180099203689080766932613397, −2.40984997072479603826990605551, −2.01513183981443905366231857112, −1.59768821872599677791134235527, −1.59312683230284723648540780305, −1.47307077394530054384548780923, −1.37982334156225288983331223639, −0.873282736670917357803788238461, −0.67698163222371326710688079041, −0.48334758071664142102092831429, −0.21897402390584219998668592611, 0.21897402390584219998668592611, 0.48334758071664142102092831429, 0.67698163222371326710688079041, 0.873282736670917357803788238461, 1.37982334156225288983331223639, 1.47307077394530054384548780923, 1.59312683230284723648540780305, 1.59768821872599677791134235527, 2.01513183981443905366231857112, 2.40984997072479603826990605551, 2.44180099203689080766932613397, 2.52434041632518813850684728423, 2.57422703179546698489388327359, 2.86875618377608157782837725252, 3.04722589507455425534693724351, 3.16200537274495010437109373348, 3.16723053330200251047595758924, 3.21696094288303624487013342666, 3.54494205123792899073369566456, 3.55892345482151924433836672681, 3.59688393453420917764806683674, 3.85083996976866560884685640932, 3.86848390508924113478945668033, 4.16690213423016237680055152272, 4.27819902713521196060327028478
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.