Dirichlet series
| L(s) = 1 | + 2-s − 4·3-s − 5·4-s − 8·5-s − 4·6-s − 8·8-s + 3·9-s − 8·10-s + 11-s + 20·12-s + 13-s + 32·15-s + 8·16-s − 22·17-s + 3·18-s − 8·19-s + 40·20-s + 22-s + 9·23-s + 32·24-s + 36·25-s + 26-s + 13·27-s + 8·29-s + 32·30-s − 3·31-s + 23·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 2.30·3-s − 5/2·4-s − 3.57·5-s − 1.63·6-s − 2.82·8-s + 9-s − 2.52·10-s + 0.301·11-s + 5.77·12-s + 0.277·13-s + 8.26·15-s + 2·16-s − 5.33·17-s + 0.707·18-s − 1.83·19-s + 8.94·20-s + 0.213·22-s + 1.87·23-s + 6.53·24-s + 36/5·25-s + 0.196·26-s + 2.50·27-s + 1.48·29-s + 5.84·30-s − 0.538·31-s + 4.06·32-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{16} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 7^{16} \cdot 29^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(5^{8} \cdot 7^{16} \cdot 29^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.07332\times 10^{14}\) |
| Root analytic conductor: | \(7.53217\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 5^{8} \cdot 7^{16} \cdot 29^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 5 | \( ( 1 + T )^{8} \) |
| 7 | \( 1 \) | |
| 29 | \( ( 1 - T )^{8} \) | |
| good | 2 | \( 1 - T + 3 p T^{2} - 3 T^{3} + 17 T^{4} + T^{5} + p^{5} T^{6} + 19 T^{7} + 15 p^{2} T^{8} + 19 p T^{9} + p^{7} T^{10} + p^{3} T^{11} + 17 p^{4} T^{12} - 3 p^{5} T^{13} + 3 p^{7} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) |
| 3 | \( 1 + 4 T + 13 T^{2} + p^{3} T^{3} + 49 T^{4} + 41 T^{5} - T^{6} - 170 T^{7} - 332 T^{8} - 170 p T^{9} - p^{2} T^{10} + 41 p^{3} T^{11} + 49 p^{4} T^{12} + p^{8} T^{13} + 13 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 - T + 34 T^{2} + 52 T^{3} + 486 T^{4} + 1930 T^{5} + 8134 T^{6} + 22657 T^{7} + 122242 T^{8} + 22657 p T^{9} + 8134 p^{2} T^{10} + 1930 p^{3} T^{11} + 486 p^{4} T^{12} + 52 p^{5} T^{13} + 34 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - T + 3 p T^{2} - 154 T^{3} + 966 T^{4} - 4284 T^{5} + 21985 T^{6} - 73349 T^{7} + 341426 T^{8} - 73349 p T^{9} + 21985 p^{2} T^{10} - 4284 p^{3} T^{11} + 966 p^{4} T^{12} - 154 p^{5} T^{13} + 3 p^{7} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 22 T + 311 T^{2} + 3153 T^{3} + 1516 p T^{4} + 174289 T^{5} + 1011921 T^{6} + 5080680 T^{7} + 22395510 T^{8} + 5080680 p T^{9} + 1011921 p^{2} T^{10} + 174289 p^{3} T^{11} + 1516 p^{5} T^{12} + 3153 p^{5} T^{13} + 311 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 8 T + 112 T^{2} + 30 p T^{3} + 5071 T^{4} + 20684 T^{5} + 154764 T^{6} + 554210 T^{7} + 3490676 T^{8} + 554210 p T^{9} + 154764 p^{2} T^{10} + 20684 p^{3} T^{11} + 5071 p^{4} T^{12} + 30 p^{6} T^{13} + 112 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 9 T + 85 T^{2} - 549 T^{3} + 3803 T^{4} - 19897 T^{5} + 112775 T^{6} - 522157 T^{7} + 2718800 T^{8} - 522157 p T^{9} + 112775 p^{2} T^{10} - 19897 p^{3} T^{11} + 3803 p^{4} T^{12} - 549 p^{5} T^{13} + 85 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 3 T + 154 T^{2} + 540 T^{3} + 11782 T^{4} + 44142 T^{5} + 591390 T^{6} + 2136089 T^{7} + 21353394 T^{8} + 2136089 p T^{9} + 591390 p^{2} T^{10} + 44142 p^{3} T^{11} + 11782 p^{4} T^{12} + 540 p^{5} T^{13} + 154 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 9 T + 172 T^{2} - 1080 T^{3} + 11964 T^{4} - 69702 T^{5} + 620556 T^{6} - 3769585 T^{7} + 26963342 T^{8} - 3769585 p T^{9} + 620556 p^{2} T^{10} - 69702 p^{3} T^{11} + 11964 p^{4} T^{12} - 1080 p^{5} T^{13} + 172 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 24 T + 363 T^{2} + 4313 T^{3} + 44231 T^{4} + 398051 T^{5} + 3239077 T^{6} + 23693584 T^{7} + 158415236 T^{8} + 23693584 p T^{9} + 3239077 p^{2} T^{10} + 398051 p^{3} T^{11} + 44231 p^{4} T^{12} + 4313 p^{5} T^{13} + 363 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 - 7 T + 242 T^{2} - 1362 T^{3} + 27408 T^{4} - 132432 T^{5} + 1990590 T^{6} - 8356197 T^{7} + 101407974 T^{8} - 8356197 p T^{9} + 1990590 p^{2} T^{10} - 132432 p^{3} T^{11} + 27408 p^{4} T^{12} - 1362 p^{5} T^{13} + 242 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 21 T + 405 T^{2} + 5561 T^{3} + 68109 T^{4} + 692041 T^{5} + 6391967 T^{6} + 51225429 T^{7} + 374522572 T^{8} + 51225429 p T^{9} + 6391967 p^{2} T^{10} + 692041 p^{3} T^{11} + 68109 p^{4} T^{12} + 5561 p^{5} T^{13} + 405 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 2 T + 228 T^{2} + 620 T^{3} + 26255 T^{4} + 90088 T^{5} + 2069984 T^{6} + 141162 p T^{7} + 123926280 T^{8} + 141162 p^{2} T^{9} + 2069984 p^{2} T^{10} + 90088 p^{3} T^{11} + 26255 p^{4} T^{12} + 620 p^{5} T^{13} + 228 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 18 T + 335 T^{2} + 3035 T^{3} + 34030 T^{4} + 236865 T^{5} + 46699 p T^{6} + 20176508 T^{7} + 205339106 T^{8} + 20176508 p T^{9} + 46699 p^{3} T^{10} + 236865 p^{3} T^{11} + 34030 p^{4} T^{12} + 3035 p^{5} T^{13} + 335 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 26 T + 604 T^{2} + 9542 T^{3} + 134592 T^{4} + 1560090 T^{5} + 16461700 T^{6} + 149841974 T^{7} + 1253627038 T^{8} + 149841974 p T^{9} + 16461700 p^{2} T^{10} + 1560090 p^{3} T^{11} + 134592 p^{4} T^{12} + 9542 p^{5} T^{13} + 604 p^{6} T^{14} + 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 8 T + 267 T^{2} + 1387 T^{3} + 34791 T^{4} + 147617 T^{5} + 3243401 T^{6} + 11483394 T^{7} + 235496232 T^{8} + 11483394 p T^{9} + 3243401 p^{2} T^{10} + 147617 p^{3} T^{11} + 34791 p^{4} T^{12} + 1387 p^{5} T^{13} + 267 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 12 T + 277 T^{2} + 3335 T^{3} + 45747 T^{4} + 472265 T^{5} + 5120167 T^{6} + 46563014 T^{7} + 413475088 T^{8} + 46563014 p T^{9} + 5120167 p^{2} T^{10} + 472265 p^{3} T^{11} + 45747 p^{4} T^{12} + 3335 p^{5} T^{13} + 277 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - T + 357 T^{2} - 787 T^{3} + 61917 T^{4} - 196949 T^{5} + 6968839 T^{6} - 25018839 T^{7} + 579523100 T^{8} - 25018839 p T^{9} + 6968839 p^{2} T^{10} - 196949 p^{3} T^{11} + 61917 p^{4} T^{12} - 787 p^{5} T^{13} + 357 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 14 T + 355 T^{2} + 4177 T^{3} + 68236 T^{4} + 685095 T^{5} + 8517781 T^{6} + 75086948 T^{7} + 784024822 T^{8} + 75086948 p T^{9} + 8517781 p^{2} T^{10} + 685095 p^{3} T^{11} + 68236 p^{4} T^{12} + 4177 p^{5} T^{13} + 355 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 10 T + 539 T^{2} - 3673 T^{3} + 121144 T^{4} - 553035 T^{5} + 15992005 T^{6} - 51259488 T^{7} + 1506945054 T^{8} - 51259488 p T^{9} + 15992005 p^{2} T^{10} - 553035 p^{3} T^{11} + 121144 p^{4} T^{12} - 3673 p^{5} T^{13} + 539 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 4 T + 437 T^{2} - 949 T^{3} + 91765 T^{4} - 99987 T^{5} + 12811723 T^{6} - 8274848 T^{7} + 1317737584 T^{8} - 8274848 p T^{9} + 12811723 p^{2} T^{10} - 99987 p^{3} T^{11} + 91765 p^{4} T^{12} - 949 p^{5} T^{13} + 437 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 11 T + 613 T^{2} - 5373 T^{3} + 164759 T^{4} - 1179335 T^{5} + 26683215 T^{6} - 160637713 T^{7} + 3017814904 T^{8} - 160637713 p T^{9} + 26683215 p^{2} T^{10} - 1179335 p^{3} T^{11} + 164759 p^{4} T^{12} - 5373 p^{5} T^{13} + 613 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.66605701774550700757023638248, −3.43831921671319697923391668570, −3.38877672442148903011889942247, −3.36861369350623621286187030110, −3.36312535569617953731643737377, −3.17962872377194462214262437439, −3.11252323291366390725809240384, −3.08331919209685413254757342555, −2.93905735315992319201820646827, −2.72420806100616242600409257425, −2.59470648907501329163702963227, −2.49789389017988314942539595668, −2.47314405672338139434684510705, −2.32184418283286602177675087733, −1.90606153369517402767992457590, −1.90430240952916275034085484689, −1.89830211943779983784184746700, −1.81553639954435665206336605845, −1.75542971322059426231305810688, −1.17968304368599453986336931370, −1.14176201453417181116973853537, −0.980377263654519216301931863761, −0.973789116672353320713693627701, −0.936635164900566428408503180134, −0.64998111119099945606506686300, 0, 0, 0, 0, 0, 0, 0, 0, 0.64998111119099945606506686300, 0.936635164900566428408503180134, 0.973789116672353320713693627701, 0.980377263654519216301931863761, 1.14176201453417181116973853537, 1.17968304368599453986336931370, 1.75542971322059426231305810688, 1.81553639954435665206336605845, 1.89830211943779983784184746700, 1.90430240952916275034085484689, 1.90606153369517402767992457590, 2.32184418283286602177675087733, 2.47314405672338139434684510705, 2.49789389017988314942539595668, 2.59470648907501329163702963227, 2.72420806100616242600409257425, 2.93905735315992319201820646827, 3.08331919209685413254757342555, 3.11252323291366390725809240384, 3.17962872377194462214262437439, 3.36312535569617953731643737377, 3.36861369350623621286187030110, 3.38877672442148903011889942247, 3.43831921671319697923391668570, 3.66605701774550700757023638248
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.