Dirichlet series
| L(s) = 1 | − 2·2-s + 3·4-s − 2·5-s + 9·7-s + 8-s + 4·10-s + 3·11-s + 10·13-s − 18·14-s − 16-s + 2·17-s − 2·19-s − 6·20-s − 6·22-s + 2·23-s + 25-s − 20·26-s + 27·28-s − 14·29-s − 5·31-s + 7·32-s − 4·34-s − 18·35-s − 27·37-s + 4·38-s − 2·40-s − 41-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s − 0.894·5-s + 3.40·7-s + 0.353·8-s + 1.26·10-s + 0.904·11-s + 2.77·13-s − 4.81·14-s − 1/4·16-s + 0.485·17-s − 0.458·19-s − 1.34·20-s − 1.27·22-s + 0.417·23-s + 1/5·25-s − 3.92·26-s + 5.10·28-s − 2.59·29-s − 0.898·31-s + 1.23·32-s − 0.685·34-s − 3.04·35-s − 4.43·37-s + 0.648·38-s − 0.316·40-s − 0.156·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{16} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(3^{16} \cdot 5^{8} \cdot 11^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(59574.2\) |
| Root analytic conductor: | \(1.98811\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 3^{16} \cdot 5^{8} \cdot 11^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(5.124679621\) |
| \(L(\frac12)\) | \(\approx\) | \(5.124679621\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 3 | \( 1 \) |
| 5 | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) | |
| 11 | \( 1 - 3 T + 8 T^{2} - 51 T^{3} + 265 T^{4} - 51 p T^{5} + 8 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) | |
| good | 2 | \( 1 + p T + T^{2} - 5 T^{3} - 7 p T^{4} - 19 T^{5} - 5 T^{6} + 41 T^{7} + 87 T^{8} + 41 p T^{9} - 5 p^{2} T^{10} - 19 p^{3} T^{11} - 7 p^{5} T^{12} - 5 p^{5} T^{13} + p^{6} T^{14} + p^{8} T^{15} + p^{8} T^{16} \) |
| 7 | \( 1 - 9 T + 41 T^{2} - 152 T^{3} + 73 p T^{4} - 1632 T^{5} + 5270 T^{6} - 2235 p T^{7} + 42197 T^{8} - 2235 p^{2} T^{9} + 5270 p^{2} T^{10} - 1632 p^{3} T^{11} + 73 p^{5} T^{12} - 152 p^{5} T^{13} + 41 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 13 | \( 1 - 10 T + 31 T^{2} + 80 T^{3} - 888 T^{4} + 2270 T^{5} + 1523 T^{6} - 23760 T^{7} + 90635 T^{8} - 23760 p T^{9} + 1523 p^{2} T^{10} + 2270 p^{3} T^{11} - 888 p^{4} T^{12} + 80 p^{5} T^{13} + 31 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 2 T - 13 T^{2} + 6 T^{3} - 142 T^{4} + 2404 T^{5} + 5259 T^{6} - 29742 T^{7} - 20357 T^{8} - 29742 p T^{9} + 5259 p^{2} T^{10} + 2404 p^{3} T^{11} - 142 p^{4} T^{12} + 6 p^{5} T^{13} - 13 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 19 | \( 1 + 2 T - 37 T^{2} - 28 T^{3} + 756 T^{4} + 470 T^{5} - 4053 T^{6} + 660 T^{7} - 25493 T^{8} + 660 p T^{9} - 4053 p^{2} T^{10} + 470 p^{3} T^{11} + 756 p^{4} T^{12} - 28 p^{5} T^{13} - 37 p^{6} T^{14} + 2 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 - T + 53 T^{2} - 40 T^{3} + 1721 T^{4} - 40 p T^{5} + 53 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 + 14 T + 177 T^{2} + 1538 T^{3} + 12792 T^{4} + 88140 T^{5} + 588283 T^{6} + 3400816 T^{7} + 19500135 T^{8} + 3400816 p T^{9} + 588283 p^{2} T^{10} + 88140 p^{3} T^{11} + 12792 p^{4} T^{12} + 1538 p^{5} T^{13} + 177 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( 1 + 5 T + 33 T^{2} + 405 T^{3} + 1803 T^{4} + 12330 T^{5} + 89286 T^{6} + 403200 T^{7} + 2179305 T^{8} + 403200 p T^{9} + 89286 p^{2} T^{10} + 12330 p^{3} T^{11} + 1803 p^{4} T^{12} + 405 p^{5} T^{13} + 33 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 37 | \( 1 + 27 T + 341 T^{2} + 3195 T^{3} + 28221 T^{4} + 221856 T^{5} + 1507630 T^{6} + 9928356 T^{7} + 63369287 T^{8} + 9928356 p T^{9} + 1507630 p^{2} T^{10} + 221856 p^{3} T^{11} + 28221 p^{4} T^{12} + 3195 p^{5} T^{13} + 341 p^{6} T^{14} + 27 p^{7} T^{15} + p^{8} T^{16} \) | |
| 41 | \( 1 + T - 37 T^{2} - 254 T^{3} + 2517 T^{4} - 1400 T^{5} - 103108 T^{6} + 463 T^{7} + 8109675 T^{8} + 463 p T^{9} - 103108 p^{2} T^{10} - 1400 p^{3} T^{11} + 2517 p^{4} T^{12} - 254 p^{5} T^{13} - 37 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( ( 1 + 14 T + 222 T^{2} + 1863 T^{3} + 15403 T^{4} + 1863 p T^{5} + 222 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 47 | \( 1 - 27 T + 7 p T^{2} - 2472 T^{3} + 12983 T^{4} - 35552 T^{5} - 232694 T^{6} + 4362679 T^{7} - 37118887 T^{8} + 4362679 p T^{9} - 232694 p^{2} T^{10} - 35552 p^{3} T^{11} + 12983 p^{4} T^{12} - 2472 p^{5} T^{13} + 7 p^{7} T^{14} - 27 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 - T - 56 T^{2} + 60 T^{3} + 1306 T^{4} - 9803 T^{5} + 73180 T^{6} + 241938 T^{7} - 7253673 T^{8} + 241938 p T^{9} + 73180 p^{2} T^{10} - 9803 p^{3} T^{11} + 1306 p^{4} T^{12} + 60 p^{5} T^{13} - 56 p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 + 13 T - 15 T^{2} - 845 T^{3} - 835 T^{4} + 25564 T^{5} - 81578 T^{6} - 1006440 T^{7} + 1157705 T^{8} - 1006440 p T^{9} - 81578 p^{2} T^{10} + 25564 p^{3} T^{11} - 835 p^{4} T^{12} - 845 p^{5} T^{13} - 15 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 3 T - 12 T^{2} - 11 T^{3} + 4632 T^{4} + 4840 T^{5} - 169898 T^{6} - 446772 T^{7} + 11653975 T^{8} - 446772 p T^{9} - 169898 p^{2} T^{10} + 4840 p^{3} T^{11} + 4632 p^{4} T^{12} - 11 p^{5} T^{13} - 12 p^{6} T^{14} + 3 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( ( 1 - 5 T + 194 T^{2} - 825 T^{3} + 18067 T^{4} - 825 p T^{5} + 194 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 71 | \( 1 + 9 T + 127 T^{2} + 891 T^{3} + 17211 T^{4} + 68910 T^{5} + 709868 T^{6} + 1927200 T^{7} + 72267317 T^{8} + 1927200 p T^{9} + 709868 p^{2} T^{10} + 68910 p^{3} T^{11} + 17211 p^{4} T^{12} + 891 p^{5} T^{13} + 127 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 - 5 T + 172 T^{2} - 1015 T^{3} + 18770 T^{4} - 198530 T^{5} + 1678892 T^{6} - 20313760 T^{7} + 111467959 T^{8} - 20313760 p T^{9} + 1678892 p^{2} T^{10} - 198530 p^{3} T^{11} + 18770 p^{4} T^{12} - 1015 p^{5} T^{13} + 172 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 10 T + 249 T^{2} + 1350 T^{3} + 31905 T^{4} + 113670 T^{5} + 2915691 T^{6} + 5002890 T^{7} + 225768564 T^{8} + 5002890 p T^{9} + 2915691 p^{2} T^{10} + 113670 p^{3} T^{11} + 31905 p^{4} T^{12} + 1350 p^{5} T^{13} + 249 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( 1 - 25 T + 169 T^{2} + 445 T^{3} - 7103 T^{4} + 51200 T^{5} - 1168898 T^{6} + 6729200 T^{7} + 8445705 T^{8} + 6729200 p T^{9} - 1168898 p^{2} T^{10} + 51200 p^{3} T^{11} - 7103 p^{4} T^{12} + 445 p^{5} T^{13} + 169 p^{6} T^{14} - 25 p^{7} T^{15} + p^{8} T^{16} \) | |
| 89 | \( ( 1 + 2 T + 301 T^{2} + 378 T^{3} + 37835 T^{4} + 378 p T^{5} + 301 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 97 | \( 1 - 13 T - 34 T^{2} + 1820 T^{3} - 7434 T^{4} + 81731 T^{5} - 1025100 T^{6} - 12168234 T^{7} + 351506887 T^{8} - 12168234 p T^{9} - 1025100 p^{2} T^{10} + 81731 p^{3} T^{11} - 7434 p^{4} T^{12} + 1820 p^{5} T^{13} - 34 p^{6} T^{14} - 13 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.82832720507628684417630966033, −4.59414973100661421883609689863, −4.55155579355944116642839284562, −4.46482602893316131620616247060, −4.06148558656043988505000441499, −4.05490175091307024709467566771, −4.02856901375325205931718585099, −3.74569847112449302276945602516, −3.71708286460596791817010228893, −3.56167176808713498923545687008, −3.37847048359737975806204199152, −3.34318020522954078711427735897, −3.02827887911275659557565395993, −2.86483001152841644750447164537, −2.70172654243589276824799674684, −2.12042858164013661905221116434, −1.91508442905222973308036018387, −1.88741093335984853616543246860, −1.86184190349788574265475505569, −1.78681515686238012040615522687, −1.51530909531972264778653161731, −1.27725115677054592444774553769, −1.16469299189460043830268697416, −0.857944538188016119850375289328, −0.46228619832269552148873043920, 0.46228619832269552148873043920, 0.857944538188016119850375289328, 1.16469299189460043830268697416, 1.27725115677054592444774553769, 1.51530909531972264778653161731, 1.78681515686238012040615522687, 1.86184190349788574265475505569, 1.88741093335984853616543246860, 1.91508442905222973308036018387, 2.12042858164013661905221116434, 2.70172654243589276824799674684, 2.86483001152841644750447164537, 3.02827887911275659557565395993, 3.34318020522954078711427735897, 3.37847048359737975806204199152, 3.56167176808713498923545687008, 3.71708286460596791817010228893, 3.74569847112449302276945602516, 4.02856901375325205931718585099, 4.05490175091307024709467566771, 4.06148558656043988505000441499, 4.46482602893316131620616247060, 4.55155579355944116642839284562, 4.59414973100661421883609689863, 4.82832720507628684417630966033
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.