Dirichlet series
| L(s) = 1 | + 4·2-s + 8·4-s − 6·5-s + 2·7-s + 16·8-s − 3·9-s − 24·10-s + 18·11-s + 8·14-s + 36·16-s − 42·17-s − 12·18-s − 46·19-s − 48·20-s + 72·22-s − 36·23-s + 18·25-s + 24·27-s + 16·28-s − 6·29-s − 32·31-s + 64·32-s − 168·34-s − 12·35-s − 24·36-s + 106·37-s − 184·38-s + ⋯ |
| L(s) = 1 | + 2·2-s + 2·4-s − 6/5·5-s + 2/7·7-s + 2·8-s − 1/3·9-s − 2.39·10-s + 1.63·11-s + 4/7·14-s + 9/4·16-s − 2.47·17-s − 2/3·18-s − 2.42·19-s − 2.39·20-s + 3.27·22-s − 1.56·23-s + 0.719·25-s + 8/9·27-s + 4/7·28-s − 0.206·29-s − 1.03·31-s + 2·32-s − 4.94·34-s − 0.342·35-s − 2/3·36-s + 2.86·37-s − 4.84·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 13^{16}\) |
| Sign: | $1$ |
| Analytic conductor: | \(5.17623\times 10^{7}\) |
| Root analytic conductor: | \(3.03477\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 13^{16} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.4635724043\) |
| \(L(\frac12)\) | \(\approx\) | \(0.4635724043\) |
| \(L(2)\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( 1 \) | |
| good | 3 | \( 1 + p T^{2} - 8 p T^{3} - 29 p T^{4} + 8 p^{2} T^{5} - 2 p^{3} T^{6} + 200 p^{2} T^{7} + 70 p^{2} T^{8} + 200 p^{4} T^{9} - 2 p^{7} T^{10} + 8 p^{8} T^{11} - 29 p^{9} T^{12} - 8 p^{11} T^{13} + p^{13} T^{14} + p^{16} T^{16} \) |
| 5 | \( 1 + 6 T + 18 T^{2} + 48 p T^{3} + 89 p T^{4} - 2988 T^{5} + 2862 T^{6} - 36666 T^{7} - 716556 T^{8} - 36666 p^{2} T^{9} + 2862 p^{4} T^{10} - 2988 p^{6} T^{11} + 89 p^{9} T^{12} + 48 p^{11} T^{13} + 18 p^{12} T^{14} + 6 p^{14} T^{15} + p^{16} T^{16} \) | |
| 7 | \( 1 - 2 T - 127 T^{2} + 62 p T^{3} + 8255 T^{4} - 22156 T^{5} - 385248 T^{6} + 453708 T^{7} + 19028482 T^{8} + 453708 p^{2} T^{9} - 385248 p^{4} T^{10} - 22156 p^{6} T^{11} + 8255 p^{8} T^{12} + 62 p^{11} T^{13} - 127 p^{12} T^{14} - 2 p^{14} T^{15} + p^{16} T^{16} \) | |
| 11 | \( 1 - 18 T + 105 T^{2} + 3114 T^{3} - 59393 T^{4} + 565908 T^{5} + 897792 T^{6} - 64810956 T^{7} + 1103342370 T^{8} - 64810956 p^{2} T^{9} + 897792 p^{4} T^{10} + 565908 p^{6} T^{11} - 59393 p^{8} T^{12} + 3114 p^{10} T^{13} + 105 p^{12} T^{14} - 18 p^{14} T^{15} + p^{16} T^{16} \) | |
| 17 | \( 1 + 42 T + 1726 T^{2} + 47796 T^{3} + 1303465 T^{4} + 30227796 T^{5} + 646060522 T^{6} + 12441458526 T^{7} + 218725843324 T^{8} + 12441458526 p^{2} T^{9} + 646060522 p^{4} T^{10} + 30227796 p^{6} T^{11} + 1303465 p^{8} T^{12} + 47796 p^{10} T^{13} + 1726 p^{12} T^{14} + 42 p^{14} T^{15} + p^{16} T^{16} \) | |
| 19 | \( 1 + 46 T + 977 T^{2} + 19730 T^{3} + 483311 T^{4} + 8783612 T^{5} + 6242688 p T^{6} + 1680839364 T^{7} + 28608116338 T^{8} + 1680839364 p^{2} T^{9} + 6242688 p^{5} T^{10} + 8783612 p^{6} T^{11} + 483311 p^{8} T^{12} + 19730 p^{10} T^{13} + 977 p^{12} T^{14} + 46 p^{14} T^{15} + p^{16} T^{16} \) | |
| 23 | \( 1 + 36 T + 2131 T^{2} + 61164 T^{3} + 2152429 T^{4} + 44226000 T^{5} + 1303450510 T^{6} + 22452294672 T^{7} + 648751140166 T^{8} + 22452294672 p^{2} T^{9} + 1303450510 p^{4} T^{10} + 44226000 p^{6} T^{11} + 2152429 p^{8} T^{12} + 61164 p^{10} T^{13} + 2131 p^{12} T^{14} + 36 p^{14} T^{15} + p^{16} T^{16} \) | |
| 29 | \( 1 + 6 T - 2398 T^{2} - 14772 T^{3} + 2997253 T^{4} + 14035500 T^{5} - 3325399546 T^{6} - 4535559294 T^{7} + 3230740374436 T^{8} - 4535559294 p^{2} T^{9} - 3325399546 p^{4} T^{10} + 14035500 p^{6} T^{11} + 2997253 p^{8} T^{12} - 14772 p^{10} T^{13} - 2398 p^{12} T^{14} + 6 p^{14} T^{15} + p^{16} T^{16} \) | |
| 31 | \( 1 + 32 T + 512 T^{2} + 24592 T^{3} + 1881968 T^{4} + 48317872 T^{5} + 884987520 T^{6} + 50977297056 T^{7} + 2940963189598 T^{8} + 50977297056 p^{2} T^{9} + 884987520 p^{4} T^{10} + 48317872 p^{6} T^{11} + 1881968 p^{8} T^{12} + 24592 p^{10} T^{13} + 512 p^{12} T^{14} + 32 p^{14} T^{15} + p^{16} T^{16} \) | |
| 37 | \( 1 - 106 T + 8342 T^{2} - 393068 T^{3} + 15188585 T^{4} - 276977564 T^{5} - 1726711374 T^{6} + 651456022674 T^{7} - 28349233687076 T^{8} + 651456022674 p^{2} T^{9} - 1726711374 p^{4} T^{10} - 276977564 p^{6} T^{11} + 15188585 p^{8} T^{12} - 393068 p^{10} T^{13} + 8342 p^{12} T^{14} - 106 p^{14} T^{15} + p^{16} T^{16} \) | |
| 41 | \( 1 + 132 T + 10686 T^{2} + 13512 p T^{3} + 19988317 T^{4} + 213687144 T^{5} - 26569998546 T^{6} - 2518966024308 T^{7} - 124070043071700 T^{8} - 2518966024308 p^{2} T^{9} - 26569998546 p^{4} T^{10} + 213687144 p^{6} T^{11} + 19988317 p^{8} T^{12} + 13512 p^{11} T^{13} + 10686 p^{12} T^{14} + 132 p^{14} T^{15} + p^{16} T^{16} \) | |
| 43 | \( 1 + 108 T + 253 p T^{2} + 755028 T^{3} + 48271669 T^{4} + 2548436904 T^{5} + 130030600102 T^{6} + 5877290089008 T^{7} + 263654368009582 T^{8} + 5877290089008 p^{2} T^{9} + 130030600102 p^{4} T^{10} + 2548436904 p^{6} T^{11} + 48271669 p^{8} T^{12} + 755028 p^{10} T^{13} + 253 p^{13} T^{14} + 108 p^{14} T^{15} + p^{16} T^{16} \) | |
| 47 | \( 1 + 60 T + 1800 T^{2} + 154932 T^{3} + 21951088 T^{4} + 867532140 T^{5} + 24541932312 T^{6} + 2003115348324 T^{7} + 163148147612766 T^{8} + 2003115348324 p^{2} T^{9} + 24541932312 p^{4} T^{10} + 867532140 p^{6} T^{11} + 21951088 p^{8} T^{12} + 154932 p^{10} T^{13} + 1800 p^{12} T^{14} + 60 p^{14} T^{15} + p^{16} T^{16} \) | |
| 53 | \( ( 1 + 66 T + 5917 T^{2} + 184818 T^{3} + 12226368 T^{4} + 184818 p^{2} T^{5} + 5917 p^{4} T^{6} + 66 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 59 | \( 1 + 18 T + 5565 T^{2} + 318294 T^{3} + 23254759 T^{4} + 1693659348 T^{5} + 98133474288 T^{6} + 8885846915100 T^{7} + 327983542866810 T^{8} + 8885846915100 p^{2} T^{9} + 98133474288 p^{4} T^{10} + 1693659348 p^{6} T^{11} + 23254759 p^{8} T^{12} + 318294 p^{10} T^{13} + 5565 p^{12} T^{14} + 18 p^{14} T^{15} + p^{16} T^{16} \) | |
| 61 | \( 1 - 36 T - 11878 T^{2} + 285336 T^{3} + 88972381 T^{4} - 1311266160 T^{5} - 475974647662 T^{6} + 1883345164380 T^{7} + 2030181383859340 T^{8} + 1883345164380 p^{2} T^{9} - 475974647662 p^{4} T^{10} - 1311266160 p^{6} T^{11} + 88972381 p^{8} T^{12} + 285336 p^{10} T^{13} - 11878 p^{12} T^{14} - 36 p^{14} T^{15} + p^{16} T^{16} \) | |
| 67 | \( 1 - 74 T + 15917 T^{2} - 695782 T^{3} + 97615703 T^{4} - 493359916 T^{5} + 227553513504 T^{6} + 21342931471668 T^{7} + 178591314167674 T^{8} + 21342931471668 p^{2} T^{9} + 227553513504 p^{4} T^{10} - 493359916 p^{6} T^{11} + 97615703 p^{8} T^{12} - 695782 p^{10} T^{13} + 15917 p^{12} T^{14} - 74 p^{14} T^{15} + p^{16} T^{16} \) | |
| 71 | \( 1 - 174 T + 14793 T^{2} - 561450 T^{3} - 2330369 T^{4} + 584834196 T^{5} + 83898109968 T^{6} - 14951669162460 T^{7} + 1250020614182514 T^{8} - 14951669162460 p^{2} T^{9} + 83898109968 p^{4} T^{10} + 584834196 p^{6} T^{11} - 2330369 p^{8} T^{12} - 561450 p^{10} T^{13} + 14793 p^{12} T^{14} - 174 p^{14} T^{15} + p^{16} T^{16} \) | |
| 73 | \( 1 + 166 T + 13778 T^{2} + 1402664 T^{3} + 172017437 T^{4} + 14047512068 T^{5} + 945563904750 T^{6} + 1139077533150 p T^{7} + 1353169547476 p^{2} T^{8} + 1139077533150 p^{3} T^{9} + 945563904750 p^{4} T^{10} + 14047512068 p^{6} T^{11} + 172017437 p^{8} T^{12} + 1402664 p^{10} T^{13} + 13778 p^{12} T^{14} + 166 p^{14} T^{15} + p^{16} T^{16} \) | |
| 79 | \( ( 1 + 48 T + 10084 T^{2} + 724368 T^{3} + 50280774 T^{4} + 724368 p^{2} T^{5} + 10084 p^{4} T^{6} + 48 p^{6} T^{7} + p^{8} T^{8} )^{2} \) | |
| 83 | \( 1 - 240 T + 28800 T^{2} - 2896512 T^{3} + 298194640 T^{4} - 338245056 p T^{5} + 2344726766592 T^{6} - 177350340390480 T^{7} + 13560551099314782 T^{8} - 177350340390480 p^{2} T^{9} + 2344726766592 p^{4} T^{10} - 338245056 p^{7} T^{11} + 298194640 p^{8} T^{12} - 2896512 p^{10} T^{13} + 28800 p^{12} T^{14} - 240 p^{14} T^{15} + p^{16} T^{16} \) | |
| 89 | \( 1 + 294 T + 63735 T^{2} + 10176306 T^{3} + 1386303445 T^{4} + 163006886868 T^{5} + 17292939386730 T^{6} + 1689166620371832 T^{7} + 154185854226666234 T^{8} + 1689166620371832 p^{2} T^{9} + 17292939386730 p^{4} T^{10} + 163006886868 p^{6} T^{11} + 1386303445 p^{8} T^{12} + 10176306 p^{10} T^{13} + 63735 p^{12} T^{14} + 294 p^{14} T^{15} + p^{16} T^{16} \) | |
| 97 | \( 1 - 58 T + 14363 T^{2} - 998 T^{3} - 4472611 T^{4} + 1301881588 T^{5} - 229071912798 T^{6} - 178692170625312 T^{7} + 7600464823934194 T^{8} - 178692170625312 p^{2} T^{9} - 229071912798 p^{4} T^{10} + 1301881588 p^{6} T^{11} - 4472611 p^{8} T^{12} - 998 p^{10} T^{13} + 14363 p^{12} T^{14} - 58 p^{14} T^{15} + p^{16} T^{16} \) | |
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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
−4.89109159094025517936410540829, −4.83760807640780919287842459222, −4.71739768217818180166834929099, −4.35745712705453702725545737494, −4.32555936344864970085321760751, −4.00978048062683762189656117665, −3.98049905053855099247269054542, −3.86485899714211385261287708114, −3.84001022726869775001076809555, −3.71125723060810259611836252628, −3.51345044579596127100929366842, −3.14034784907551724413766161724, −3.11384702129993471370110328438, −3.08931537498911507821379373417, −2.42668256900254986110215382141, −2.35567760704635370127419068692, −2.31402789570292293874918267980, −2.31031174975688494621728106373, −1.86700888988663367760680948934, −1.77589494520834256985733992196, −1.35524752111151555228759170266, −1.17639835970534213350890204504, −1.11342776329874020525479235339, −0.15237096669055170877194689380, −0.14972034804234306090631538744, 0.14972034804234306090631538744, 0.15237096669055170877194689380, 1.11342776329874020525479235339, 1.17639835970534213350890204504, 1.35524752111151555228759170266, 1.77589494520834256985733992196, 1.86700888988663367760680948934, 2.31031174975688494621728106373, 2.31402789570292293874918267980, 2.35567760704635370127419068692, 2.42668256900254986110215382141, 3.08931537498911507821379373417, 3.11384702129993471370110328438, 3.14034784907551724413766161724, 3.51345044579596127100929366842, 3.71125723060810259611836252628, 3.84001022726869775001076809555, 3.86485899714211385261287708114, 3.98049905053855099247269054542, 4.00978048062683762189656117665, 4.32555936344864970085321760751, 4.35745712705453702725545737494, 4.71739768217818180166834929099, 4.83760807640780919287842459222, 4.89109159094025517936410540829
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.