Dirichlet series
| L(s) = 1 | + 3-s − 4·5-s + 4·7-s + 6·9-s − 8·11-s + 13-s − 4·15-s + 13·17-s + 19-s + 4·21-s + 8·23-s + 6·25-s + 27-s − 3·29-s + 4·31-s − 8·33-s − 16·35-s + 20·37-s + 39-s − 8·41-s − 3·43-s − 24·45-s + 10·47-s − 14·49-s + 13·51-s − 11·53-s + 32·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s + 1.51·7-s + 2·9-s − 2.41·11-s + 0.277·13-s − 1.03·15-s + 3.15·17-s + 0.229·19-s + 0.872·21-s + 1.66·23-s + 6/5·25-s + 0.192·27-s − 0.557·29-s + 0.718·31-s − 1.39·33-s − 2.70·35-s + 3.28·37-s + 0.160·39-s − 1.24·41-s − 0.457·43-s − 3.57·45-s + 1.45·47-s − 2·49-s + 1.82·51-s − 1.51·53-s + 4.31·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{24} \cdot 5^{8} \cdot 19^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.83960\times 10^{6}\) |
| Root analytic conductor: | \(2.46345\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{24} \cdot 5^{8} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.6695234399\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6695234399\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 + T + T^{2} )^{4} \) | |
| 19 | \( 1 - T + p T^{2} + 151 T^{3} - 169 T^{4} + 151 p T^{5} + p^{3} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 3 | \( 1 - T - 5 T^{2} + 10 T^{3} + 4 T^{4} - 8 p T^{5} + 4 T^{6} + 5 p T^{7} + 11 p T^{8} + 5 p^{2} T^{9} + 4 p^{2} T^{10} - 8 p^{4} T^{11} + 4 p^{4} T^{12} + 10 p^{5} T^{13} - 5 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 2 T + 13 T^{2} - 19 T^{3} + 96 T^{4} - 19 p T^{5} + 13 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 11 | \( ( 1 + 4 T + 29 T^{2} + 123 T^{3} + 423 T^{4} + 123 p T^{5} + 29 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 13 | \( 1 - T - 38 T^{2} + 53 T^{3} + 787 T^{4} - 1016 T^{5} - 12132 T^{6} + 6208 T^{7} + 164620 T^{8} + 6208 p T^{9} - 12132 p^{2} T^{10} - 1016 p^{3} T^{11} + 787 p^{4} T^{12} + 53 p^{5} T^{13} - 38 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 13 T + 3 p T^{2} - 62 T^{3} + 886 T^{4} - 9140 T^{5} + 37784 T^{6} - 108283 T^{7} + 385105 T^{8} - 108283 p T^{9} + 37784 p^{2} T^{10} - 9140 p^{3} T^{11} + 886 p^{4} T^{12} - 62 p^{5} T^{13} + 3 p^{7} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( 1 - 8 T + 21 T^{2} - 26 T^{3} - 177 T^{4} + 1570 T^{5} + 9097 T^{6} - 84696 T^{7} + 286467 T^{8} - 84696 p T^{9} + 9097 p^{2} T^{10} + 1570 p^{3} T^{11} - 177 p^{4} T^{12} - 26 p^{5} T^{13} + 21 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 + 3 T - 76 T^{2} - 47 T^{3} + 3613 T^{4} - 60 p T^{5} - 115881 T^{6} + 20201 T^{7} + 2973548 T^{8} + 20201 p T^{9} - 115881 p^{2} T^{10} - 60 p^{4} T^{11} + 3613 p^{4} T^{12} - 47 p^{5} T^{13} - 76 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( ( 1 - 2 T + 85 T^{2} - 283 T^{3} + 3279 T^{4} - 283 p T^{5} + 85 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 37 | \( ( 1 - 10 T + 113 T^{2} - 673 T^{3} + 5278 T^{4} - 673 p T^{5} + 113 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 41 | \( 1 + 8 T - 105 T^{2} - 446 T^{3} + 11039 T^{4} + 23510 T^{5} - 664263 T^{6} - 347592 T^{7} + 31557745 T^{8} - 347592 p T^{9} - 664263 p^{2} T^{10} + 23510 p^{3} T^{11} + 11039 p^{4} T^{12} - 446 p^{5} T^{13} - 105 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 3 T - 107 T^{2} - 500 T^{3} + 5868 T^{4} + 29520 T^{5} - 178608 T^{6} - 669867 T^{7} + 5293669 T^{8} - 669867 p T^{9} - 178608 p^{2} T^{10} + 29520 p^{3} T^{11} + 5868 p^{4} T^{12} - 500 p^{5} T^{13} - 107 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 10 T - 63 T^{2} + 1040 T^{3} + 2283 T^{4} - 61630 T^{5} + 92341 T^{6} + 1216410 T^{7} - 6974481 T^{8} + 1216410 p T^{9} + 92341 p^{2} T^{10} - 61630 p^{3} T^{11} + 2283 p^{4} T^{12} + 1040 p^{5} T^{13} - 63 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 11 T - 49 T^{2} - 466 T^{3} + 4268 T^{4} + 1216 T^{5} - 364940 T^{6} - 393171 T^{7} + 13516437 T^{8} - 393171 p T^{9} - 364940 p^{2} T^{10} + 1216 p^{3} T^{11} + 4268 p^{4} T^{12} - 466 p^{5} T^{13} - 49 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - T - 150 T^{2} - 697 T^{3} + 12859 T^{4} + 77760 T^{5} - 400567 T^{6} - 2806629 T^{7} + 8872130 T^{8} - 2806629 p T^{9} - 400567 p^{2} T^{10} + 77760 p^{3} T^{11} + 12859 p^{4} T^{12} - 697 p^{5} T^{13} - 150 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 170 T^{2} + 400 T^{3} + 15633 T^{4} - 46200 T^{5} - 950250 T^{6} + 1494600 T^{7} + 52190548 T^{8} + 1494600 p T^{9} - 950250 p^{2} T^{10} - 46200 p^{3} T^{11} + 15633 p^{4} T^{12} + 400 p^{5} T^{13} - 170 p^{6} T^{14} + p^{8} T^{16} \) | |
| 67 | \( 1 + 8 T - 200 T^{2} - 1024 T^{3} + 30722 T^{4} + 96128 T^{5} - 2989152 T^{6} - 2140456 T^{7} + 239298291 T^{8} - 2140456 p T^{9} - 2989152 p^{2} T^{10} + 96128 p^{3} T^{11} + 30722 p^{4} T^{12} - 1024 p^{5} T^{13} - 200 p^{6} T^{14} + 8 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 + 4 T - 113 T^{2} + 250 T^{3} + 5432 T^{4} - 58816 T^{5} - 96838 T^{6} + 3016547 T^{7} + 14896600 T^{8} + 3016547 p T^{9} - 96838 p^{2} T^{10} - 58816 p^{3} T^{11} + 5432 p^{4} T^{12} + 250 p^{5} T^{13} - 113 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 20 T + T^{2} - 994 T^{3} + 22127 T^{4} + 224504 T^{5} - 1051575 T^{6} + 1307720 T^{7} + 205219449 T^{8} + 1307720 p T^{9} - 1051575 p^{2} T^{10} + 224504 p^{3} T^{11} + 22127 p^{4} T^{12} - 994 p^{5} T^{13} + p^{6} T^{14} + 20 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 3 T - 135 T^{2} - 1190 T^{3} + 3481 T^{4} + 91921 T^{5} + 128554 T^{6} - 2256001 T^{7} + 8242970 T^{8} - 2256001 p T^{9} + 128554 p^{2} T^{10} + 91921 p^{3} T^{11} + 3481 p^{4} T^{12} - 1190 p^{5} T^{13} - 135 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( ( 1 + 15 T + 276 T^{2} + 2312 T^{3} + 28404 T^{4} + 2312 p T^{5} + 276 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 - 17 T + 75 T^{2} + 2168 T^{3} - 34549 T^{4} + 174187 T^{5} + 792642 T^{6} - 20195577 T^{7} + 192568360 T^{8} - 20195577 p T^{9} + 792642 p^{2} T^{10} + 174187 p^{3} T^{11} - 34549 p^{4} T^{12} + 2168 p^{5} T^{13} + 75 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 - 11 T - 209 T^{2} + 1434 T^{3} + 36108 T^{4} - 92484 T^{5} - 4910880 T^{6} + 4116487 T^{7} + 505431757 T^{8} + 4116487 p T^{9} - 4910880 p^{2} T^{10} - 92484 p^{3} T^{11} + 36108 p^{4} T^{12} + 1434 p^{5} T^{13} - 209 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.57582986030935619475670363591, −4.33129298366950069027021702796, −4.26490867757936288665456250046, −4.06419472693393362217808766003, −3.91859668231606565691893828015, −3.90108656151089728725805652452, −3.57779237357833421933888189701, −3.48412697621826824174185516061, −3.32680509967531593206291900077, −3.18962978011461707343742185291, −3.17824804643201823759064044138, −2.96585443405253634704878720392, −2.82914592383782865748777191270, −2.63509180059583115298022630848, −2.52949183714569162733480049551, −2.45274691473987810945664736513, −1.92152161919928797601117453211, −1.90062819870407155793676971019, −1.63606301629739767067938242280, −1.52270152551230507011378489111, −1.44373222818674297066029506109, −0.989747457857108387346557723790, −0.989251596735325955379014634779, −0.75379024652668632679098345037, −0.096390207384453015183770465998, 0.096390207384453015183770465998, 0.75379024652668632679098345037, 0.989251596735325955379014634779, 0.989747457857108387346557723790, 1.44373222818674297066029506109, 1.52270152551230507011378489111, 1.63606301629739767067938242280, 1.90062819870407155793676971019, 1.92152161919928797601117453211, 2.45274691473987810945664736513, 2.52949183714569162733480049551, 2.63509180059583115298022630848, 2.82914592383782865748777191270, 2.96585443405253634704878720392, 3.17824804643201823759064044138, 3.18962978011461707343742185291, 3.32680509967531593206291900077, 3.48412697621826824174185516061, 3.57779237357833421933888189701, 3.90108656151089728725805652452, 3.91859668231606565691893828015, 4.06419472693393362217808766003, 4.26490867757936288665456250046, 4.33129298366950069027021702796, 4.57582986030935619475670363591
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.