Dirichlet series
| L(s) = 1 | + 8·3-s + 5-s + 36·9-s − 5·11-s + 8·15-s − 7·17-s − 24·19-s − 23-s − 18·25-s + 120·27-s − 11·29-s − 30·31-s − 40·33-s + 11·37-s + 10·41-s − 24·43-s + 36·45-s + 3·47-s − 25·49-s − 56·51-s − 25·53-s − 5·55-s − 192·57-s − 45·59-s + 16·61-s − 18·67-s − 8·69-s + ⋯ |
| L(s) = 1 | + 4.61·3-s + 0.447·5-s + 12·9-s − 1.50·11-s + 2.06·15-s − 1.69·17-s − 5.50·19-s − 0.208·23-s − 3.59·25-s + 23.0·27-s − 2.04·29-s − 5.38·31-s − 6.96·33-s + 1.80·37-s + 1.56·41-s − 3.65·43-s + 5.36·45-s + 0.437·47-s − 3.57·49-s − 7.84·51-s − 3.43·53-s − 0.674·55-s − 25.4·57-s − 5.85·59-s + 2.04·61-s − 2.19·67-s − 0.963·69-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 167^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 167^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{32} \cdot 3^{8} \cdot 167^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.81759\times 10^{14}\) |
| Root analytic conductor: | \(8.00050\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{32} \cdot 3^{8} \cdot 167^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
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 | 2 | \( 1 \) |
| 3 | \( ( 1 - T )^{8} \) | |
| 167 | \( ( 1 + T )^{8} \) | |
| good | 5 | \( 1 - T + 19 T^{2} - 17 T^{3} + 161 T^{4} - 192 T^{5} + 927 T^{6} - 1582 T^{7} + 4718 T^{8} - 1582 p T^{9} + 927 p^{2} T^{10} - 192 p^{3} T^{11} + 161 p^{4} T^{12} - 17 p^{5} T^{13} + 19 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 + 25 T^{2} - p T^{3} + 5 p^{2} T^{4} - 304 T^{5} + 157 p T^{6} - 4509 T^{7} + 3708 T^{8} - 4509 p T^{9} + 157 p^{3} T^{10} - 304 p^{3} T^{11} + 5 p^{6} T^{12} - p^{6} T^{13} + 25 p^{6} T^{14} + p^{8} T^{16} \) | |
| 11 | \( 1 + 5 T + 65 T^{2} + 281 T^{3} + 2021 T^{4} + 7532 T^{5} + 38975 T^{6} + 124436 T^{7} + 512616 T^{8} + 124436 p T^{9} + 38975 p^{2} T^{10} + 7532 p^{3} T^{11} + 2021 p^{4} T^{12} + 281 p^{5} T^{13} + 65 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 + 43 T^{2} - 10 T^{3} + 1062 T^{4} - 887 T^{5} + 18051 T^{6} - 22315 T^{7} + 251414 T^{8} - 22315 p T^{9} + 18051 p^{2} T^{10} - 887 p^{3} T^{11} + 1062 p^{4} T^{12} - 10 p^{5} T^{13} + 43 p^{6} T^{14} + p^{8} T^{16} \) | |
| 17 | \( 1 + 7 T + 94 T^{2} + 515 T^{3} + 3983 T^{4} + 19188 T^{5} + 110490 T^{6} + 27950 p T^{7} + 2210030 T^{8} + 27950 p^{2} T^{9} + 110490 p^{2} T^{10} + 19188 p^{3} T^{11} + 3983 p^{4} T^{12} + 515 p^{5} T^{13} + 94 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 24 T + 353 T^{2} + 3710 T^{3} + 31022 T^{4} + 214835 T^{5} + 1278735 T^{6} + 6653837 T^{7} + 30755986 T^{8} + 6653837 p T^{9} + 1278735 p^{2} T^{10} + 214835 p^{3} T^{11} + 31022 p^{4} T^{12} + 3710 p^{5} T^{13} + 353 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 + T + 3 p T^{2} - 65 T^{3} + 3165 T^{4} - 2232 T^{5} + 113855 T^{6} - 101490 T^{7} + 2806012 T^{8} - 101490 p T^{9} + 113855 p^{2} T^{10} - 2232 p^{3} T^{11} + 3165 p^{4} T^{12} - 65 p^{5} T^{13} + 3 p^{7} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 11 T + 179 T^{2} + 1685 T^{3} + 15871 T^{4} + 120326 T^{5} + 863835 T^{6} + 5257502 T^{7} + 30730100 T^{8} + 5257502 p T^{9} + 863835 p^{2} T^{10} + 120326 p^{3} T^{11} + 15871 p^{4} T^{12} + 1685 p^{5} T^{13} + 179 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 30 T + 494 T^{2} + 5767 T^{3} + 53036 T^{4} + 409236 T^{5} + 2761738 T^{6} + 16906143 T^{7} + 96763478 T^{8} + 16906143 p T^{9} + 2761738 p^{2} T^{10} + 409236 p^{3} T^{11} + 53036 p^{4} T^{12} + 5767 p^{5} T^{13} + 494 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - 11 T + 189 T^{2} - 1649 T^{3} + 17551 T^{4} - 135404 T^{5} + 1078209 T^{6} - 7208484 T^{7} + 46667124 T^{8} - 7208484 p T^{9} + 1078209 p^{2} T^{10} - 135404 p^{3} T^{11} + 17551 p^{4} T^{12} - 1649 p^{5} T^{13} + 189 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 10 T + 232 T^{2} - 1869 T^{3} + 25858 T^{4} - 173932 T^{5} + 1795868 T^{6} - 10291247 T^{7} + 87043884 T^{8} - 10291247 p T^{9} + 1795868 p^{2} T^{10} - 173932 p^{3} T^{11} + 25858 p^{4} T^{12} - 1869 p^{5} T^{13} + 232 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 24 T + 480 T^{2} + 6573 T^{3} + 79642 T^{4} + 786764 T^{5} + 7047644 T^{6} + 54159389 T^{7} + 381070708 T^{8} + 54159389 p T^{9} + 7047644 p^{2} T^{10} + 786764 p^{3} T^{11} + 79642 p^{4} T^{12} + 6573 p^{5} T^{13} + 480 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 3 T + 73 T^{2} - 628 T^{3} + 6578 T^{4} - 33029 T^{5} + 475607 T^{6} - 2491768 T^{7} + 21399806 T^{8} - 2491768 p T^{9} + 475607 p^{2} T^{10} - 33029 p^{3} T^{11} + 6578 p^{4} T^{12} - 628 p^{5} T^{13} + 73 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 25 T + 579 T^{2} + 8774 T^{3} + 120610 T^{4} + 1329957 T^{5} + 13375065 T^{6} + 114505786 T^{7} + 896980008 T^{8} + 114505786 p T^{9} + 13375065 p^{2} T^{10} + 1329957 p^{3} T^{11} + 120610 p^{4} T^{12} + 8774 p^{5} T^{13} + 579 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 45 T + 1223 T^{2} + 23998 T^{3} + 374940 T^{4} + 4846851 T^{5} + 53304093 T^{6} + 504932174 T^{7} + 4157961070 T^{8} + 504932174 p T^{9} + 53304093 p^{2} T^{10} + 4846851 p^{3} T^{11} + 374940 p^{4} T^{12} + 23998 p^{5} T^{13} + 1223 p^{6} T^{14} + 45 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 16 T + 449 T^{2} - 5808 T^{3} + 91382 T^{4} - 952529 T^{5} + 10887925 T^{6} - 91603311 T^{7} + 824334746 T^{8} - 91603311 p T^{9} + 10887925 p^{2} T^{10} - 952529 p^{3} T^{11} + 91382 p^{4} T^{12} - 5808 p^{5} T^{13} + 449 p^{6} T^{14} - 16 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 18 T + 475 T^{2} + 6870 T^{3} + 105772 T^{4} + 1204725 T^{5} + 13774321 T^{6} + 126193127 T^{7} + 1141351048 T^{8} + 126193127 p T^{9} + 13774321 p^{2} T^{10} + 1204725 p^{3} T^{11} + 105772 p^{4} T^{12} + 6870 p^{5} T^{13} + 475 p^{6} T^{14} + 18 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 21 T + 408 T^{2} + 4842 T^{3} + 53937 T^{4} + 469505 T^{5} + 3864512 T^{6} + 29321392 T^{7} + 233979356 T^{8} + 29321392 p T^{9} + 3864512 p^{2} T^{10} + 469505 p^{3} T^{11} + 53937 p^{4} T^{12} + 4842 p^{5} T^{13} + 408 p^{6} T^{14} + 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 8 T + 361 T^{2} + 2398 T^{3} + 62862 T^{4} + 365975 T^{5} + 7401469 T^{6} + 38062427 T^{7} + 634690566 T^{8} + 38062427 p T^{9} + 7401469 p^{2} T^{10} + 365975 p^{3} T^{11} + 62862 p^{4} T^{12} + 2398 p^{5} T^{13} + 361 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 10 T + 266 T^{2} + 1825 T^{3} + 37702 T^{4} + 229058 T^{5} + 4155060 T^{6} + 23409469 T^{7} + 372961876 T^{8} + 23409469 p T^{9} + 4155060 p^{2} T^{10} + 229058 p^{3} T^{11} + 37702 p^{4} T^{12} + 1825 p^{5} T^{13} + 266 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 7 T + 397 T^{2} + 3500 T^{3} + 78910 T^{4} + 711159 T^{5} + 10765451 T^{6} + 84640014 T^{7} + 1058869618 T^{8} + 84640014 p T^{9} + 10765451 p^{2} T^{10} + 711159 p^{3} T^{11} + 78910 p^{4} T^{12} + 3500 p^{5} T^{13} + 397 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 - 26 T + 761 T^{2} - 12182 T^{3} + 203638 T^{4} - 2354455 T^{5} + 29224825 T^{6} - 274376689 T^{7} + 2923563118 T^{8} - 274376689 p T^{9} + 29224825 p^{2} T^{10} - 2354455 p^{3} T^{11} + 203638 p^{4} T^{12} - 12182 p^{5} T^{13} + 761 p^{6} T^{14} - 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 3 T + 402 T^{2} + 2024 T^{3} + 73933 T^{4} + 595959 T^{5} + 8542284 T^{6} + 96913774 T^{7} + 827247384 T^{8} + 96913774 p T^{9} + 8542284 p^{2} T^{10} + 595959 p^{3} T^{11} + 73933 p^{4} T^{12} + 2024 p^{5} T^{13} + 402 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−3.58704657495712227534352906135, −3.31230503834023189195231836984, −3.27379107741395648074016779125, −3.22336427473509524651197851604, −3.12113088083305001299666012938, −3.09151293373972892395310155155, −3.02360104539158246870156380918, −2.91279365404630935159018986483, −2.64912829251183526818614912466, −2.39169029208986507564480009199, −2.39063952106361361154224130143, −2.33326938488455236613261176594, −2.32663144974998468172919944471, −2.25843420312732843714400661492, −2.20544410073396800348871648139, −2.15149606833635619728267220300, −2.01484290660410208911895311810, −1.78798190345569241025000424516, −1.51439200578073031517558226092, −1.46182295332302988660850216447, −1.44515229452879826290113240883, −1.40651131417006142443635112465, −1.40527607006618159850014428401, −1.39093313080999075357337994497, −1.22071990757236067502057754502, 0, 0, 0, 0, 0, 0, 0, 0, 1.22071990757236067502057754502, 1.39093313080999075357337994497, 1.40527607006618159850014428401, 1.40651131417006142443635112465, 1.44515229452879826290113240883, 1.46182295332302988660850216447, 1.51439200578073031517558226092, 1.78798190345569241025000424516, 2.01484290660410208911895311810, 2.15149606833635619728267220300, 2.20544410073396800348871648139, 2.25843420312732843714400661492, 2.32663144974998468172919944471, 2.33326938488455236613261176594, 2.39063952106361361154224130143, 2.39169029208986507564480009199, 2.64912829251183526818614912466, 2.91279365404630935159018986483, 3.02360104539158246870156380918, 3.09151293373972892395310155155, 3.12113088083305001299666012938, 3.22336427473509524651197851604, 3.27379107741395648074016779125, 3.31230503834023189195231836984, 3.58704657495712227534352906135
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.