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 + 36·13-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 − 144·26-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 + ⋯ |
| 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 + 2.76·13-s + 4/7·14-s + 9/4·16-s − 2.47·17-s + 2/3·18-s + 2.42·19-s + 12/5·20-s + 3.27·22-s − 1.56·23-s + 0.719·25-s − 5.53·26-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 + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0634551\) |
| Root analytic conductor: | \(0.841693\) |
| Motivic weight: | \(2\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 13^{8} ,\ ( \ : [1]^{8} ),\ 1 )\) |
Particular Values
| \(L(\frac{3}{2})\) | \(\approx\) | \(0.2493713384\) |
| \(L(\frac12)\) | \(\approx\) | \(0.2493713384\) |
| \(L(2)\) | 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 | \( 1 - 36 T + 589 T^{2} - 7824 T^{3} + 8172 p T^{4} - 7824 p^{2} T^{5} + 589 p^{4} T^{6} - 36 p^{6} T^{7} + p^{8} T^{8} \) | |
| 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
−8.508536336186558123739459273033, −8.415811805627628070892323830060, −8.414200846963753870410423786743, −8.408054940427337131476877541663, −7.938031072020424465250746126613, −7.51327801071358733594405478733, −7.32621940736378706558100165065, −7.19686918574269283431349104819, −7.00799684015701738839241920865, −6.64726784752073974216223813522, −6.59310787136014597901912063305, −6.07070787594747835733965990320, −5.86126712498175236756529226101, −5.79575268262008453379640960603, −5.68351322626094566961703263013, −5.59585237339799816418015853747, −4.73732082562542502302931571853, −4.67875794261568990535766948499, −4.51599376519828505595508846068, −3.72501728027450928871648736105, −3.34406627693961089623562420582, −3.32403950724458260739668169059, −2.45694893223072642811104559116, −2.39308500621309395742667048041, −1.40446580441834285903588825862, 1.40446580441834285903588825862, 2.39308500621309395742667048041, 2.45694893223072642811104559116, 3.32403950724458260739668169059, 3.34406627693961089623562420582, 3.72501728027450928871648736105, 4.51599376519828505595508846068, 4.67875794261568990535766948499, 4.73732082562542502302931571853, 5.59585237339799816418015853747, 5.68351322626094566961703263013, 5.79575268262008453379640960603, 5.86126712498175236756529226101, 6.07070787594747835733965990320, 6.59310787136014597901912063305, 6.64726784752073974216223813522, 7.00799684015701738839241920865, 7.19686918574269283431349104819, 7.32621940736378706558100165065, 7.51327801071358733594405478733, 7.938031072020424465250746126613, 8.408054940427337131476877541663, 8.414200846963753870410423786743, 8.415811805627628070892323830060, 8.508536336186558123739459273033
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.