Dirichlet series
| L(s) = 1 | − 3-s − 3·5-s + 7·7-s − 3·9-s + 7·11-s + 7·13-s + 3·15-s + 17-s − 2·19-s − 7·21-s − 4·23-s − 7·25-s − 3·27-s + 17·29-s − 13·31-s − 7·33-s − 21·35-s + 37-s − 7·39-s + 9·41-s + 6·43-s + 9·45-s − 47-s + 33·49-s − 51-s − 33·53-s − 21·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.34·5-s + 2.64·7-s − 9-s + 2.11·11-s + 1.94·13-s + 0.774·15-s + 0.242·17-s − 0.458·19-s − 1.52·21-s − 0.834·23-s − 7/5·25-s − 0.577·27-s + 3.15·29-s − 2.33·31-s − 1.21·33-s − 3.54·35-s + 0.164·37-s − 1.12·39-s + 1.40·41-s + 0.914·43-s + 1.34·45-s − 0.145·47-s + 33/7·49-s − 0.140·51-s − 4.53·53-s − 2.83·55-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{24} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(0.0594399\) |
| Root analytic conductor: | \(0.838262\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{24} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(0.6667849386\) |
| \(L(\frac12)\) | \(\approx\) | \(0.6667849386\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 - 7 T + 23 T^{2} - 9 p T^{3} + 40 p T^{4} - 9 p^{2} T^{5} + 23 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 3 | \( 1 + T + 4 T^{2} + 10 T^{3} + 16 T^{4} + 53 T^{5} + 95 T^{6} + 44 p T^{7} + 352 T^{8} + 44 p^{2} T^{9} + 95 p^{2} T^{10} + 53 p^{3} T^{11} + 16 p^{4} T^{12} + 10 p^{5} T^{13} + 4 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
| 5 | \( 1 + 3 T + 16 T^{2} + 34 T^{3} + 134 T^{4} + 237 T^{5} + 819 T^{6} + 1386 T^{7} + 4556 T^{8} + 1386 p T^{9} + 819 p^{2} T^{10} + 237 p^{3} T^{11} + 134 p^{4} T^{12} + 34 p^{5} T^{13} + 16 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 7 | \( 1 - p T + 16 T^{2} - 20 T^{3} + 76 T^{4} - 341 T^{5} + 1045 T^{6} - 1576 T^{7} + 1052 T^{8} - 1576 p T^{9} + 1045 p^{2} T^{10} - 341 p^{3} T^{11} + 76 p^{4} T^{12} - 20 p^{5} T^{13} + 16 p^{6} T^{14} - p^{8} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 7 T - 6 T^{2} + 96 T^{3} + 76 T^{4} - 1831 T^{5} + 9105 T^{6} - 1150 T^{7} - 119328 T^{8} - 1150 p T^{9} + 9105 p^{2} T^{10} - 1831 p^{3} T^{11} + 76 p^{4} T^{12} + 96 p^{5} T^{13} - 6 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - T - 12 T^{2} + 4 p T^{3} - 124 T^{4} - 1871 T^{5} + 7215 T^{6} + 17704 T^{7} - 121576 T^{8} + 17704 p T^{9} + 7215 p^{2} T^{10} - 1871 p^{3} T^{11} - 124 p^{4} T^{12} + 4 p^{6} T^{13} - 12 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( ( 1 + T - 3 T^{2} - 67 T^{3} + 140 T^{4} - 67 p T^{5} - 3 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 23 | \( ( 1 + 2 T + 40 T^{2} + 10 T^{3} + 718 T^{4} + 10 p T^{5} + 40 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 - 17 T + 72 T^{2} + 272 T^{3} - 2708 T^{4} + 5425 T^{5} + 2763 T^{6} - 218004 T^{7} + 2125200 T^{8} - 218004 p T^{9} + 2763 p^{2} T^{10} + 5425 p^{3} T^{11} - 2708 p^{4} T^{12} + 272 p^{5} T^{13} + 72 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 13 T + 34 T^{2} - 220 T^{3} - 1102 T^{4} - 5365 T^{5} - 55879 T^{6} + 86932 T^{7} + 2801428 T^{8} + 86932 p T^{9} - 55879 p^{2} T^{10} - 5365 p^{3} T^{11} - 1102 p^{4} T^{12} - 220 p^{5} T^{13} + 34 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - T + 36 T^{2} + 282 T^{3} + 1726 T^{4} + 5717 T^{5} + 72975 T^{6} + 560030 T^{7} + 399612 T^{8} + 560030 p T^{9} + 72975 p^{2} T^{10} + 5717 p^{3} T^{11} + 1726 p^{4} T^{12} + 282 p^{5} T^{13} + 36 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 - 9 T - 68 T^{2} + 964 T^{3} - 724 T^{4} - 42615 T^{5} + 290343 T^{6} + 684120 T^{7} - 16705288 T^{8} + 684120 p T^{9} + 290343 p^{2} T^{10} - 42615 p^{3} T^{11} - 724 p^{4} T^{12} + 964 p^{5} T^{13} - 68 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 - 3 T + 157 T^{2} - 343 T^{3} + 9788 T^{4} - 343 p T^{5} + 157 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 + T - 82 T^{2} - 298 T^{3} + 58 p T^{4} + 19841 T^{5} + 46575 T^{6} - 509444 T^{7} - 6186936 T^{8} - 509444 p T^{9} + 46575 p^{2} T^{10} + 19841 p^{3} T^{11} + 58 p^{5} T^{12} - 298 p^{5} T^{13} - 82 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 33 T + 430 T^{2} + 2882 T^{3} + 11534 T^{4} + 17589 T^{5} - 162291 T^{6} + 814044 T^{7} + 23430476 T^{8} + 814044 p T^{9} - 162291 p^{2} T^{10} + 17589 p^{3} T^{11} + 11534 p^{4} T^{12} + 2882 p^{5} T^{13} + 430 p^{6} T^{14} + 33 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( ( 1 + 15 T + 41 T^{2} - 45 T^{3} + 1156 T^{4} - 45 p T^{5} + 41 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 61 | \( 1 + 9 T + 42 T^{2} + 6 p T^{3} - 94 T^{4} - 18015 T^{5} + 207033 T^{6} + 3524700 T^{7} + 24465892 T^{8} + 3524700 p T^{9} + 207033 p^{2} T^{10} - 18015 p^{3} T^{11} - 94 p^{4} T^{12} + 6 p^{6} T^{13} + 42 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 + 5 T + 167 T^{2} + 1265 T^{3} + 13224 T^{4} + 1265 p T^{5} + 167 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 + 25 T + 260 T^{2} + 1900 T^{3} + 15914 T^{4} + 158945 T^{5} + 1541235 T^{6} + 11895270 T^{7} + 86952316 T^{8} + 11895270 p T^{9} + 1541235 p^{2} T^{10} + 158945 p^{3} T^{11} + 15914 p^{4} T^{12} + 1900 p^{5} T^{13} + 260 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 7 T - 16 T^{2} - 56 T^{3} + 6256 T^{4} - 18899 T^{5} - 526445 T^{6} - 2619360 T^{7} + 12491952 T^{8} - 2619360 p T^{9} - 526445 p^{2} T^{10} - 18899 p^{3} T^{11} + 6256 p^{4} T^{12} - 56 p^{5} T^{13} - 16 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + T + 46 T^{2} + 500 T^{3} + 11978 T^{4} + 75715 T^{5} + 339389 T^{6} + 7359056 T^{7} + 84903268 T^{8} + 7359056 p T^{9} + 339389 p^{2} T^{10} + 75715 p^{3} T^{11} + 11978 p^{4} T^{12} + 500 p^{5} T^{13} + 46 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 + T^{2} + 20 T^{3} + 7907 T^{4} - 6780 T^{5} - 79917 T^{6} + 872160 T^{7} + 53878160 T^{8} + 872160 p T^{9} - 79917 p^{2} T^{10} - 6780 p^{3} T^{11} + 7907 p^{4} T^{12} + 20 p^{5} T^{13} + p^{6} T^{14} + p^{8} T^{16} \) | |
| 89 | \( ( 1 - 11 T + 383 T^{2} - 2901 T^{3} + 52208 T^{4} - 2901 p T^{5} + 383 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - 8 T - 183 T^{2} + 644 T^{3} + 17703 T^{4} - 58924 T^{5} + 1087779 T^{6} - 2218448 T^{7} - 205877032 T^{8} - 2218448 p T^{9} + 1087779 p^{2} T^{10} - 58924 p^{3} T^{11} + 17703 p^{4} T^{12} + 644 p^{5} T^{13} - 183 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−6.63515881102461809257291131879, −6.34556284675936029893112443808, −6.32512830560935829903076658066, −6.30286830873058049496413269708, −5.92967155195243747467799717630, −5.86062674911274810946932579818, −5.84078559867491675572117746424, −5.65494677619223767135687256230, −5.38496339913248008121413490930, −5.01197158605453838713700572981, −4.88248786800161012562497863830, −4.58679618610465481788523530638, −4.38892839022517179518003360939, −4.30769204199739065159331135535, −4.19875310245102798135588155225, −4.15691103043512354354772451617, −3.75485069214614195799214868475, −3.50160437216349931700718841341, −3.07034682258235743759936609826, −3.05937822435631955990361179325, −2.96470734760702677030878070680, −1.98093926901652019296961263951, −1.87329240694976561342095209592, −1.57955015345605611159126836536, −1.33737819644955968517654449962, 1.33737819644955968517654449962, 1.57955015345605611159126836536, 1.87329240694976561342095209592, 1.98093926901652019296961263951, 2.96470734760702677030878070680, 3.05937822435631955990361179325, 3.07034682258235743759936609826, 3.50160437216349931700718841341, 3.75485069214614195799214868475, 4.15691103043512354354772451617, 4.19875310245102798135588155225, 4.30769204199739065159331135535, 4.38892839022517179518003360939, 4.58679618610465481788523530638, 4.88248786800161012562497863830, 5.01197158605453838713700572981, 5.38496339913248008121413490930, 5.65494677619223767135687256230, 5.84078559867491675572117746424, 5.86062674911274810946932579818, 5.92967155195243747467799717630, 6.30286830873058049496413269708, 6.32512830560935829903076658066, 6.34556284675936029893112443808, 6.63515881102461809257291131879
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.