Dirichlet series
| L(s) = 1 | + 4·2-s + 8·3-s + 6·4-s − 2·5-s + 32·6-s + 3·7-s + 36·9-s − 8·10-s + 4·11-s + 48·12-s + 7·13-s + 12·14-s − 16·15-s − 15·16-s − 6·17-s + 144·18-s − 4·19-s − 12·20-s + 24·21-s + 16·22-s + 4·23-s + 15·25-s + 28·26-s + 120·27-s + 18·28-s + 6·29-s − 64·30-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4.61·3-s + 3·4-s − 0.894·5-s + 13.0·6-s + 1.13·7-s + 12·9-s − 2.52·10-s + 1.20·11-s + 13.8·12-s + 1.94·13-s + 3.20·14-s − 4.13·15-s − 3.75·16-s − 1.45·17-s + 33.9·18-s − 0.917·19-s − 2.68·20-s + 5.23·21-s + 3.41·22-s + 0.834·23-s + 3·25-s + 5.49·26-s + 23.0·27-s + 3.40·28-s + 1.11·29-s − 11.6·30-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(130544.\) |
| Root analytic conductor: | \(2.08802\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 3^{8} \cdot 7^{8} \cdot 13^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(190.6675844\) |
| \(L(\frac12)\) | \(\approx\) | \(190.6675844\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} )^{4} \) |
| 3 | \( ( 1 - T )^{8} \) | |
| 7 | \( 1 - 3 T - 4 T^{2} + 3 T^{3} + 57 T^{4} + 3 p T^{5} - 4 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | |
| 13 | \( 1 - 7 T + 22 T^{2} - 5 p T^{3} + 17 p T^{4} - 5 p^{2} T^{5} + 22 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 5 | \( ( 1 + T - 6 T^{2} - 3 T^{3} + 19 T^{4} - 3 p T^{5} - 6 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 2 T + 8 T^{2} - 3 T^{3} - 39 T^{4} - 3 p T^{5} + 8 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 17 | \( 1 + 6 T - 2 p T^{2} - 150 T^{3} + 1379 T^{4} + 3063 T^{5} - 34155 T^{6} - 11307 T^{7} + 753011 T^{8} - 11307 p T^{9} - 34155 p^{2} T^{10} + 3063 p^{3} T^{11} + 1379 p^{4} T^{12} - 150 p^{5} T^{13} - 2 p^{7} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( ( 1 + 2 T + 25 T^{2} + 10 T^{3} + 527 T^{4} + 10 p T^{5} + 25 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 23 | \( 1 - 4 T - 64 T^{2} + 142 T^{3} + 2881 T^{4} - 3033 T^{5} - 90851 T^{6} + 32687 T^{7} + 2230625 T^{8} + 32687 p T^{9} - 90851 p^{2} T^{10} - 3033 p^{3} T^{11} + 2881 p^{4} T^{12} + 142 p^{5} T^{13} - 64 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 29 | \( 1 - 6 T - 82 T^{2} + 294 T^{3} + 5747 T^{4} - 11595 T^{5} - 247875 T^{6} + 96411 T^{7} + 8798855 T^{8} + 96411 p T^{9} - 247875 p^{2} T^{10} - 11595 p^{3} T^{11} + 5747 p^{4} T^{12} + 294 p^{5} T^{13} - 82 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 10 T - 50 T^{2} + 370 T^{3} + 5775 T^{4} - 21735 T^{5} - 233575 T^{6} + 60465 T^{7} + 10701829 T^{8} + 60465 p T^{9} - 233575 p^{2} T^{10} - 21735 p^{3} T^{11} + 5775 p^{4} T^{12} + 370 p^{5} T^{13} - 50 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 12 T + 52 T^{2} - 6 p T^{3} - 4391 T^{4} - 32049 T^{5} - 53615 T^{6} + 853545 T^{7} + 8862835 T^{8} + 853545 p T^{9} - 53615 p^{2} T^{10} - 32049 p^{3} T^{11} - 4391 p^{4} T^{12} - 6 p^{6} T^{13} + 52 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 6 T + 2 T^{2} - 438 T^{3} - 2293 T^{4} + 5799 T^{5} + 86601 T^{6} + 350385 T^{7} - 1711837 T^{8} + 350385 p T^{9} + 86601 p^{2} T^{10} + 5799 p^{3} T^{11} - 2293 p^{4} T^{12} - 438 p^{5} T^{13} + 2 p^{6} T^{14} + 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 4 T + 12 T^{2} - 198 T^{3} - 2517 T^{4} - 11811 T^{5} - 583 T^{6} + 536039 T^{7} + 3034557 T^{8} + 536039 p T^{9} - 583 p^{2} T^{10} - 11811 p^{3} T^{11} - 2517 p^{4} T^{12} - 198 p^{5} T^{13} + 12 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 + 17 T + 119 T^{2} + 472 T^{3} + 538 T^{4} - 3153 T^{5} - 80780 T^{6} - 1729738 T^{7} - 16428613 T^{8} - 1729738 p T^{9} - 80780 p^{2} T^{10} - 3153 p^{3} T^{11} + 538 p^{4} T^{12} + 472 p^{5} T^{13} + 119 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - 3 T - 59 T^{2} - 654 T^{3} + 2656 T^{4} + 43083 T^{5} + 288700 T^{6} - 2046792 T^{7} - 13897187 T^{8} - 2046792 p T^{9} + 288700 p^{2} T^{10} + 43083 p^{3} T^{11} + 2656 p^{4} T^{12} - 654 p^{5} T^{13} - 59 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 101 T^{2} + 952 T^{4} - 230987 T^{6} + 37916959 T^{8} - 230987 p^{2} T^{10} + 952 p^{4} T^{12} - 101 p^{6} T^{14} + p^{8} T^{16} \) | |
| 61 | \( ( 1 - 4 T + 100 T^{2} - 644 T^{3} + 5606 T^{4} - 644 p T^{5} + 100 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 67 | \( ( 1 - 7 T + 163 T^{2} - 1736 T^{3} + 12551 T^{4} - 1736 p T^{5} + 163 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 - 6 T - 157 T^{2} + 486 T^{3} + 15470 T^{4} - 14844 T^{5} - 980133 T^{6} + 694554 T^{7} + 48433547 T^{8} + 694554 p T^{9} - 980133 p^{2} T^{10} - 14844 p^{3} T^{11} + 15470 p^{4} T^{12} + 486 p^{5} T^{13} - 157 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 19 T + 68 T^{2} - 1381 T^{3} - 12826 T^{4} + 30062 T^{5} + 801015 T^{6} + 1939200 T^{7} - 18767075 T^{8} + 1939200 p T^{9} + 801015 p^{2} T^{10} + 30062 p^{3} T^{11} - 12826 p^{4} T^{12} - 1381 p^{5} T^{13} + 68 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 - 24 T + 323 T^{2} - 4552 T^{3} + 47608 T^{4} - 340592 T^{5} + 2759853 T^{6} - 17950408 T^{7} + 87302199 T^{8} - 17950408 p T^{9} + 2759853 p^{2} T^{10} - 340592 p^{3} T^{11} + 47608 p^{4} T^{12} - 4552 p^{5} T^{13} + 323 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( ( 1 - 6 T + 334 T^{2} - 1473 T^{3} + 41675 T^{4} - 1473 p T^{5} + 334 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 + 7 T - 247 T^{2} - 910 T^{3} + 40576 T^{4} + 61593 T^{5} - 4851068 T^{6} - 2635640 T^{7} + 454263785 T^{8} - 2635640 p T^{9} - 4851068 p^{2} T^{10} + 61593 p^{3} T^{11} + 40576 p^{4} T^{12} - 910 p^{5} T^{13} - 247 p^{6} T^{14} + 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 25 T + 81 T^{2} - 1194 T^{3} + 24840 T^{4} + 432927 T^{5} + 137852 T^{6} + 7536320 T^{7} + 385877313 T^{8} + 7536320 p T^{9} + 137852 p^{2} T^{10} + 432927 p^{3} T^{11} + 24840 p^{4} T^{12} - 1194 p^{5} T^{13} + 81 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.60709768684538713574806093985, −4.42362704592469627942022458912, −4.42123281691841021408210298149, −4.36313093773616816404433917287, −4.12568083224652046818985613981, −4.00027971948044099575312706719, −3.74781453901634895254998468853, −3.70499929824039181065877845053, −3.57009124299413443455522262380, −3.46379615144480965538497887394, −3.42507834996319464686848945119, −3.23086291870881716671326850584, −3.16571236145940465993529137372, −2.93722833327458076350687262678, −2.62977286258131307789384477356, −2.60645717662456482595324782962, −2.48742666671619081066581039713, −2.40566558146356262592490175440, −2.14766218060726463402465887765, −1.76571510934992450011865325262, −1.70390066176149023550355274456, −1.45174955779301811774473696992, −1.27956993825752304377476436914, −1.02118599518658934194148192329, −0.68304513844620496861605701903, 0.68304513844620496861605701903, 1.02118599518658934194148192329, 1.27956993825752304377476436914, 1.45174955779301811774473696992, 1.70390066176149023550355274456, 1.76571510934992450011865325262, 2.14766218060726463402465887765, 2.40566558146356262592490175440, 2.48742666671619081066581039713, 2.60645717662456482595324782962, 2.62977286258131307789384477356, 2.93722833327458076350687262678, 3.16571236145940465993529137372, 3.23086291870881716671326850584, 3.42507834996319464686848945119, 3.46379615144480965538497887394, 3.57009124299413443455522262380, 3.70499929824039181065877845053, 3.74781453901634895254998468853, 4.00027971948044099575312706719, 4.12568083224652046818985613981, 4.36313093773616816404433917287, 4.42123281691841021408210298149, 4.42362704592469627942022458912, 4.60709768684538713574806093985
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.