Dirichlet series
| L(s) = 1 | + 8·2-s + 8·3-s + 36·4-s − 6·5-s + 64·6-s − 3·7-s + 120·8-s + 36·9-s − 48·10-s − 15·11-s + 288·12-s − 8·13-s − 24·14-s − 48·15-s + 330·16-s − 11·17-s + 288·18-s − 15·19-s − 216·20-s − 24·21-s − 120·22-s + 23-s + 960·24-s − 7·25-s − 64·26-s + 120·27-s − 108·28-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 4.61·3-s + 18·4-s − 2.68·5-s + 26.1·6-s − 1.13·7-s + 42.4·8-s + 12·9-s − 15.1·10-s − 4.52·11-s + 83.1·12-s − 2.21·13-s − 6.41·14-s − 12.3·15-s + 82.5·16-s − 2.66·17-s + 67.8·18-s − 3.44·19-s − 48.2·20-s − 5.23·21-s − 25.5·22-s + 0.208·23-s + 195.·24-s − 7/5·25-s − 12.5·26-s + 23.0·27-s − 20.4·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 13^{8} \cdot 103^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 13^{8} \cdot 103^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 13^{8} \cdot 103^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(2.86860\times 10^{14}\) |
| Root analytic conductor: | \(8.00948\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(8\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 13^{8} \cdot 103^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
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 | \( ( 1 - T )^{8} \) |
| 3 | \( ( 1 - T )^{8} \) | |
| 13 | \( ( 1 + T )^{8} \) | |
| 103 | \( ( 1 + T )^{8} \) | |
| good | 5 | \( 1 + 6 T + 43 T^{2} + 179 T^{3} + 753 T^{4} + 2414 T^{5} + 1488 p T^{6} + 3814 p T^{7} + 9246 p T^{8} + 3814 p^{2} T^{9} + 1488 p^{3} T^{10} + 2414 p^{3} T^{11} + 753 p^{4} T^{12} + 179 p^{5} T^{13} + 43 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 + 3 T + 5 p T^{2} + 82 T^{3} + 520 T^{4} + 20 p^{2} T^{5} + 661 p T^{6} + 7598 T^{7} + 33038 T^{8} + 7598 p T^{9} + 661 p^{3} T^{10} + 20 p^{5} T^{11} + 520 p^{4} T^{12} + 82 p^{5} T^{13} + 5 p^{7} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 11 | \( 1 + 15 T + 163 T^{2} + 1267 T^{3} + 8202 T^{4} + 3996 p T^{5} + 18625 p T^{6} + 823935 T^{7} + 2926889 T^{8} + 823935 p T^{9} + 18625 p^{3} T^{10} + 3996 p^{4} T^{11} + 8202 p^{4} T^{12} + 1267 p^{5} T^{13} + 163 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 11 T + 121 T^{2} + 865 T^{3} + 6128 T^{4} + 34226 T^{5} + 187199 T^{6} + 859993 T^{7} + 3847800 T^{8} + 859993 p T^{9} + 187199 p^{2} T^{10} + 34226 p^{3} T^{11} + 6128 p^{4} T^{12} + 865 p^{5} T^{13} + 121 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 15 T + 176 T^{2} + 83 p T^{3} + 11879 T^{4} + 77654 T^{5} + 446604 T^{6} + 2291803 T^{7} + 10538805 T^{8} + 2291803 p T^{9} + 446604 p^{2} T^{10} + 77654 p^{3} T^{11} + 11879 p^{4} T^{12} + 83 p^{6} T^{13} + 176 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - T + 3 p T^{2} - 79 T^{3} + 2418 T^{4} - 4870 T^{5} + 57771 T^{6} - 174833 T^{7} + 1243603 T^{8} - 174833 p T^{9} + 57771 p^{2} T^{10} - 4870 p^{3} T^{11} + 2418 p^{4} T^{12} - 79 p^{5} T^{13} + 3 p^{7} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 10 T + 132 T^{2} + 1191 T^{3} + 10091 T^{4} + 69737 T^{5} + 488836 T^{6} + 2845164 T^{7} + 16378047 T^{8} + 2845164 p T^{9} + 488836 p^{2} T^{10} + 69737 p^{3} T^{11} + 10091 p^{4} T^{12} + 1191 p^{5} T^{13} + 132 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 3 T + 140 T^{2} + 303 T^{3} + 9749 T^{4} + 17084 T^{5} + 472412 T^{6} + 743553 T^{7} + 17040609 T^{8} + 743553 p T^{9} + 472412 p^{2} T^{10} + 17084 p^{3} T^{11} + 9749 p^{4} T^{12} + 303 p^{5} T^{13} + 140 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 26 T + 534 T^{2} + 7527 T^{3} + 90799 T^{4} + 887857 T^{5} + 7663020 T^{6} + 56248822 T^{7} + 368805321 T^{8} + 56248822 p T^{9} + 7663020 p^{2} T^{10} + 887857 p^{3} T^{11} + 90799 p^{4} T^{12} + 7527 p^{5} T^{13} + 534 p^{6} T^{14} + 26 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 12 T + 263 T^{2} + 2656 T^{3} + 31913 T^{4} + 265520 T^{5} + 2357812 T^{6} + 16122221 T^{7} + 116649126 T^{8} + 16122221 p T^{9} + 2357812 p^{2} T^{10} + 265520 p^{3} T^{11} + 31913 p^{4} T^{12} + 2656 p^{5} T^{13} + 263 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 4 T + 122 T^{2} + 62 T^{3} + 9102 T^{4} + 2034 T^{5} + 597587 T^{6} + 22235 T^{7} + 28402378 T^{8} + 22235 p T^{9} + 597587 p^{2} T^{10} + 2034 p^{3} T^{11} + 9102 p^{4} T^{12} + 62 p^{5} T^{13} + 122 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 6 T + 220 T^{2} + 1226 T^{3} + 26865 T^{4} + 133374 T^{5} + 2109734 T^{6} + 9205719 T^{7} + 117697226 T^{8} + 9205719 p T^{9} + 2109734 p^{2} T^{10} + 133374 p^{3} T^{11} + 26865 p^{4} T^{12} + 1226 p^{5} T^{13} + 220 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 4 T + 360 T^{2} + 1339 T^{3} + 58863 T^{4} + 199300 T^{5} + 5761338 T^{6} + 17051412 T^{7} + 371419976 T^{8} + 17051412 p T^{9} + 5761338 p^{2} T^{10} + 199300 p^{3} T^{11} + 58863 p^{4} T^{12} + 1339 p^{5} T^{13} + 360 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 19 T + 429 T^{2} + 5392 T^{3} + 72424 T^{4} + 688510 T^{5} + 7027583 T^{6} + 55252980 T^{7} + 476866142 T^{8} + 55252980 p T^{9} + 7027583 p^{2} T^{10} + 688510 p^{3} T^{11} + 72424 p^{4} T^{12} + 5392 p^{5} T^{13} + 429 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 14 T + 314 T^{2} + 2580 T^{3} + 33268 T^{4} + 152612 T^{5} + 1680893 T^{6} + 2492193 T^{7} + 72005300 T^{8} + 2492193 p T^{9} + 1680893 p^{2} T^{10} + 152612 p^{3} T^{11} + 33268 p^{4} T^{12} + 2580 p^{5} T^{13} + 314 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 + 13 T + 346 T^{2} + 3293 T^{3} + 51717 T^{4} + 383491 T^{5} + 4693425 T^{6} + 29547058 T^{7} + 333123472 T^{8} + 29547058 p T^{9} + 4693425 p^{2} T^{10} + 383491 p^{3} T^{11} + 51717 p^{4} T^{12} + 3293 p^{5} T^{13} + 346 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 31 T + 855 T^{2} + 15308 T^{3} + 247375 T^{4} + 3153105 T^{5} + 36971771 T^{6} + 363235587 T^{7} + 3317121700 T^{8} + 363235587 p T^{9} + 36971771 p^{2} T^{10} + 3153105 p^{3} T^{11} + 247375 p^{4} T^{12} + 15308 p^{5} T^{13} + 855 p^{6} T^{14} + 31 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 27 T + 853 T^{2} + 14950 T^{3} + 268502 T^{4} + 3439427 T^{5} + 43709956 T^{6} + 428418795 T^{7} + 4126826152 T^{8} + 428418795 p T^{9} + 43709956 p^{2} T^{10} + 3439427 p^{3} T^{11} + 268502 p^{4} T^{12} + 14950 p^{5} T^{13} + 853 p^{6} T^{14} + 27 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 13 T + 537 T^{2} + 6547 T^{3} + 131837 T^{4} + 1449160 T^{5} + 19384928 T^{6} + 184019619 T^{7} + 1868852684 T^{8} + 184019619 p T^{9} + 19384928 p^{2} T^{10} + 1449160 p^{3} T^{11} + 131837 p^{4} T^{12} + 6547 p^{5} T^{13} + 537 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + 28 T + 775 T^{2} + 11623 T^{3} + 171117 T^{4} + 1590547 T^{5} + 16039995 T^{6} + 104649245 T^{7} + 1094635440 T^{8} + 104649245 p T^{9} + 16039995 p^{2} T^{10} + 1590547 p^{3} T^{11} + 171117 p^{4} T^{12} + 11623 p^{5} T^{13} + 775 p^{6} T^{14} + 28 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( 1 + 2 T + 446 T^{2} + 766 T^{3} + 97931 T^{4} + 110033 T^{5} + 13928016 T^{6} + 10186355 T^{7} + 1430090001 T^{8} + 10186355 p T^{9} + 13928016 p^{2} T^{10} + 110033 p^{3} T^{11} + 97931 p^{4} T^{12} + 766 p^{5} T^{13} + 446 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 30 T + 867 T^{2} + 14706 T^{3} + 245585 T^{4} + 2986073 T^{5} + 37779248 T^{6} + 377683163 T^{7} + 4127980869 T^{8} + 377683163 p T^{9} + 37779248 p^{2} T^{10} + 2986073 p^{3} T^{11} + 245585 p^{4} T^{12} + 14706 p^{5} T^{13} + 867 p^{6} T^{14} + 30 p^{7} T^{15} + p^{8} T^{16} \) | |
| show more | ||
| show less | ||
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.71484050643315457368494043108, −3.26518663774528227391217940842, −3.25088758180605186575832425813, −3.21713932077452656839489026328, −3.17940801305494789916247786381, −3.15487646557299542283357029707, −3.14222599464247575000522688956, −3.09370690494674986503545003999, −2.90999712712075701592147856308, −2.64576888824869862683022427214, −2.54094577105374366916033821417, −2.49245043857985550884753910335, −2.44208268529628551773802708437, −2.43441742607547720806586868812, −2.38856817870036604930731499284, −2.26115884195020724395712487097, −2.22700459171578111916171431572, −1.80628737271062788708408218186, −1.72274793057304423650960749038, −1.68022883419080830513299684911, −1.58434020733237776258913033897, −1.55756910636062427656365848545, −1.55157248212039471958871807652, −1.44725601351528442597829036255, −1.34816046001889723365295458486, 0, 0, 0, 0, 0, 0, 0, 0, 1.34816046001889723365295458486, 1.44725601351528442597829036255, 1.55157248212039471958871807652, 1.55756910636062427656365848545, 1.58434020733237776258913033897, 1.68022883419080830513299684911, 1.72274793057304423650960749038, 1.80628737271062788708408218186, 2.22700459171578111916171431572, 2.26115884195020724395712487097, 2.38856817870036604930731499284, 2.43441742607547720806586868812, 2.44208268529628551773802708437, 2.49245043857985550884753910335, 2.54094577105374366916033821417, 2.64576888824869862683022427214, 2.90999712712075701592147856308, 3.09370690494674986503545003999, 3.14222599464247575000522688956, 3.15487646557299542283357029707, 3.17940801305494789916247786381, 3.21713932077452656839489026328, 3.25088758180605186575832425813, 3.26518663774528227391217940842, 3.71484050643315457368494043108
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.