Dirichlet series
| L(s) = 1 | − 4·2-s + 2·3-s + 11·4-s − 8·6-s − 3·7-s − 23·8-s + 9-s + 3·11-s + 22·12-s + 4·13-s + 12·14-s + 39·16-s − 4·18-s + 2·19-s − 6·21-s − 12·22-s + 6·23-s − 46·24-s − 16·26-s − 33·28-s + 10·29-s + 19·31-s − 61·32-s + 6·33-s + 11·36-s + 37-s − 8·38-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 1.15·3-s + 11/2·4-s − 3.26·6-s − 1.13·7-s − 8.13·8-s + 1/3·9-s + 0.904·11-s + 6.35·12-s + 1.10·13-s + 3.20·14-s + 39/4·16-s − 0.942·18-s + 0.458·19-s − 1.30·21-s − 2.55·22-s + 1.25·23-s − 9.38·24-s − 3.13·26-s − 6.23·28-s + 1.85·29-s + 3.41·31-s − 10.7·32-s + 1.04·33-s + 11/6·36-s + 0.164·37-s − 1.29·38-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 5^{16} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{8} \cdot 5^{16} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(3.54689\times 10^{6}\) |
| Root analytic conductor: | \(2.56664\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{8} \cdot 5^{16} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(1.535883455\) |
| \(L(\frac12)\) | \(\approx\) | \(1.535883455\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 5 | \( 1 \) | |
| 11 | \( 1 - 3 T - 2 p T^{2} + 19 T^{3} + 335 T^{4} + 19 p T^{5} - 2 p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 2 | \( 1 + p^{2} T + 5 T^{2} - T^{3} - 3 p T^{4} + 7 T^{5} + 21 T^{6} - 7 T^{7} - 49 T^{8} - 7 p T^{9} + 21 p^{2} T^{10} + 7 p^{3} T^{11} - 3 p^{5} T^{12} - p^{5} T^{13} + 5 p^{6} T^{14} + p^{9} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 + 3 T - 3 T^{2} - 24 T^{3} + 3 T^{4} + 24 T^{5} - 206 T^{6} + 573 T^{7} + 5433 T^{8} + 573 p T^{9} - 206 p^{2} T^{10} + 24 p^{3} T^{11} + 3 p^{4} T^{12} - 24 p^{5} T^{13} - 3 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 4 T + 3 T^{2} - 72 T^{3} + 568 T^{4} - 124 p T^{5} + 2841 T^{6} - 23776 T^{7} + 137583 T^{8} - 23776 p T^{9} + 2841 p^{2} T^{10} - 124 p^{4} T^{11} + 568 p^{4} T^{12} - 72 p^{5} T^{13} + 3 p^{6} T^{14} - 4 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 + 13 T^{2} + 80 T^{3} + 270 T^{4} + 460 T^{5} + 5773 T^{6} + 14470 T^{7} + 23699 T^{8} + 14470 p T^{9} + 5773 p^{2} T^{10} + 460 p^{3} T^{11} + 270 p^{4} T^{12} + 80 p^{5} T^{13} + 13 p^{6} T^{14} + p^{8} T^{16} \) | |
| 19 | \( 1 - 2 T - 13 T^{2} + 72 T^{3} + 392 T^{4} + 110 T^{5} - 8697 T^{6} - 8904 T^{7} + 339075 T^{8} - 8904 p T^{9} - 8697 p^{2} T^{10} + 110 p^{3} T^{11} + 392 p^{4} T^{12} + 72 p^{5} T^{13} - 13 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 - 3 T + 37 T^{2} - 78 T^{3} + 1093 T^{4} - 78 p T^{5} + 37 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 - 10 T + 55 T^{2} - 400 T^{3} + 3024 T^{4} - 24690 T^{5} + 5055 p T^{6} - 731720 T^{7} + 4160811 T^{8} - 731720 p T^{9} + 5055 p^{3} T^{10} - 24690 p^{3} T^{11} + 3024 p^{4} T^{12} - 400 p^{5} T^{13} + 55 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 - 19 T + 173 T^{2} - 1079 T^{3} + 6207 T^{4} - 39850 T^{5} + 231342 T^{6} - 1019972 T^{7} + 4555345 T^{8} - 1019972 p T^{9} + 231342 p^{2} T^{10} - 39850 p^{3} T^{11} + 6207 p^{4} T^{12} - 1079 p^{5} T^{13} + 173 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 - T - p T^{2} + 83 T^{3} - 989 T^{4} + 8474 T^{5} + 32080 T^{6} - 331596 T^{7} + 1075119 T^{8} - 331596 p T^{9} + 32080 p^{2} T^{10} + 8474 p^{3} T^{11} - 989 p^{4} T^{12} + 83 p^{5} T^{13} - p^{7} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + 9 T + 3 p T^{2} + 974 T^{3} + 11067 T^{4} + 80950 T^{5} + 664532 T^{6} + 106557 p T^{7} + 769255 p T^{8} + 106557 p^{2} T^{9} + 664532 p^{2} T^{10} + 80950 p^{3} T^{11} + 11067 p^{4} T^{12} + 974 p^{5} T^{13} + 3 p^{7} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 + 62 T^{2} - 375 T^{3} + 1909 T^{4} - 375 p T^{5} + 62 p^{2} T^{6} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 - 19 T + 161 T^{2} - 952 T^{3} + 5681 T^{4} - 21302 T^{5} - 48990 T^{6} + 1137555 T^{7} - 8501903 T^{8} + 1137555 p T^{9} - 48990 p^{2} T^{10} - 21302 p^{3} T^{11} + 5681 p^{4} T^{12} - 952 p^{5} T^{13} + 161 p^{6} T^{14} - 19 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 25 T + 326 T^{2} + 3620 T^{3} + 38702 T^{4} + 373145 T^{5} + 3256788 T^{6} + 26526610 T^{7} + 201208695 T^{8} + 26526610 p T^{9} + 3256788 p^{2} T^{10} + 373145 p^{3} T^{11} + 38702 p^{4} T^{12} + 3620 p^{5} T^{13} + 326 p^{6} T^{14} + 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 13 T + 47 T^{2} - 47 T^{3} + 1017 T^{4} - 9820 T^{5} + 167838 T^{6} - 2814986 T^{7} + 28479065 T^{8} - 2814986 p T^{9} + 167838 p^{2} T^{10} - 9820 p^{3} T^{11} + 1017 p^{4} T^{12} - 47 p^{5} T^{13} + 47 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 - 13 T + 88 T^{2} - 659 T^{3} + 6712 T^{4} - 88400 T^{5} + 693262 T^{6} - 3514828 T^{7} + 26130695 T^{8} - 3514828 p T^{9} + 693262 p^{2} T^{10} - 88400 p^{3} T^{11} + 6712 p^{4} T^{12} - 659 p^{5} T^{13} + 88 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 + T + 168 T^{2} + 89 T^{3} + 14153 T^{4} + 89 p T^{5} + 168 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 + 11 T + 83 T^{2} + 331 T^{3} + 1907 T^{4} - 19610 T^{5} + 16782 T^{6} + 1589858 T^{7} + 30712905 T^{8} + 1589858 p T^{9} + 16782 p^{2} T^{10} - 19610 p^{3} T^{11} + 1907 p^{4} T^{12} + 331 p^{5} T^{13} + 83 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 7 T - 98 T^{2} + 469 T^{3} + 6176 T^{4} - 19502 T^{5} + 17990 T^{6} + 1098062 T^{7} - 36323381 T^{8} + 1098062 p T^{9} + 17990 p^{2} T^{10} - 19502 p^{3} T^{11} + 6176 p^{4} T^{12} + 469 p^{5} T^{13} - 98 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 22 T + 125 T^{2} - 10 T^{3} + 6225 T^{4} + 73966 T^{5} - 403353 T^{6} - 7990450 T^{7} - 58812140 T^{8} - 7990450 p T^{9} - 403353 p^{2} T^{10} + 73966 p^{3} T^{11} + 6225 p^{4} T^{12} - 10 p^{5} T^{13} + 125 p^{6} T^{14} + 22 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 21 T + 109 T^{2} - 615 T^{3} + 22511 T^{4} - 220998 T^{5} + 720980 T^{6} - 10036422 T^{7} + 165244057 T^{8} - 10036422 p T^{9} + 720980 p^{2} T^{10} - 220998 p^{3} T^{11} + 22511 p^{4} T^{12} - 615 p^{5} T^{13} + 109 p^{6} T^{14} - 21 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 10 T + 3 p T^{2} + 2360 T^{3} + 31893 T^{4} + 2360 p T^{5} + 3 p^{3} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 + 31 T + 236 T^{2} - 2512 T^{3} - 50694 T^{4} - 179387 T^{5} + 1682790 T^{6} + 13200830 T^{7} + 28675747 T^{8} + 13200830 p T^{9} + 1682790 p^{2} T^{10} - 179387 p^{3} T^{11} - 50694 p^{4} T^{12} - 2512 p^{5} T^{13} + 236 p^{6} T^{14} + 31 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.29555809799097869045394605660, −4.27455097587095417133985118543, −4.01634945441813284811754735551, −3.99799153778790712354328933976, −3.91566361282722505476714611434, −3.70389639723196781658080592606, −3.37417983498713909577887758099, −3.19962274974669881913168687593, −3.14802954771315114921889526962, −3.06703253614048790386876383523, −3.00440169890153358132491893410, −2.75357707938708358230377679900, −2.73184344621800289764851067406, −2.43742444482347952221271424747, −2.42525252715385162529797678767, −2.32546192748785917640580662200, −2.04804665410323933848652577283, −1.83961600834612299699993053991, −1.40342105232008346526579881594, −1.38275823760638317943233582535, −1.31947293072393905321363599341, −1.15711050317618743105927500844, −0.855701841355085177764351660716, −0.67965832646451745523761675502, −0.23109309044442419498346713494, 0.23109309044442419498346713494, 0.67965832646451745523761675502, 0.855701841355085177764351660716, 1.15711050317618743105927500844, 1.31947293072393905321363599341, 1.38275823760638317943233582535, 1.40342105232008346526579881594, 1.83961600834612299699993053991, 2.04804665410323933848652577283, 2.32546192748785917640580662200, 2.42525252715385162529797678767, 2.43742444482347952221271424747, 2.73184344621800289764851067406, 2.75357707938708358230377679900, 3.00440169890153358132491893410, 3.06703253614048790386876383523, 3.14802954771315114921889526962, 3.19962274974669881913168687593, 3.37417983498713909577887758099, 3.70389639723196781658080592606, 3.91566361282722505476714611434, 3.99799153778790712354328933976, 4.01634945441813284811754735551, 4.27455097587095417133985118543, 4.29555809799097869045394605660
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.