Dirichlet series
| L(s) = 1 | − 4·2-s + 3-s + 6·4-s − 4·6-s + 12·7-s + 9-s + 10·11-s + 6·12-s + 9·13-s − 48·14-s − 15·16-s + 5·17-s − 4·18-s + 12·21-s − 40·22-s + 6·23-s − 36·26-s + 8·27-s + 72·28-s + 17·29-s + 22·31-s + 24·32-s + 10·33-s − 20·34-s + 6·36-s − 8·37-s + 9·39-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 0.577·3-s + 3·4-s − 1.63·6-s + 4.53·7-s + 1/3·9-s + 3.01·11-s + 1.73·12-s + 2.49·13-s − 12.8·14-s − 3.75·16-s + 1.21·17-s − 0.942·18-s + 2.61·21-s − 8.52·22-s + 1.25·23-s − 7.06·26-s + 1.53·27-s + 13.6·28-s + 3.15·29-s + 3.95·31-s + 4.24·32-s + 1.74·33-s − 3.42·34-s + 36-s − 1.31·37-s + 1.44·39-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 5^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(16\) |
| Conductor: | \(2^{8} \cdot 5^{16} \cdot 19^{8}\) |
| Sign: | $1$ |
| Analytic conductor: | \(1.09649\times 10^{7}\) |
| Root analytic conductor: | \(2.75423\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((16,\ 2^{8} \cdot 5^{16} \cdot 19^{8} ,\ ( \ : [1/2]^{8} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(13.76273540\) |
| \(L(\frac12)\) | \(\approx\) | \(13.76273540\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 + T + T^{2} )^{4} \) |
| 5 | \( 1 \) | |
| 19 | \( 1 + 35 T^{2} + 36 p T^{4} + 35 p^{2} T^{6} + p^{4} T^{8} \) | |
| good | 3 | \( 1 - T - 7 T^{3} + 8 T^{4} + 4 T^{5} + 40 T^{6} - 22 p T^{7} - 47 T^{8} - 22 p^{2} T^{9} + 40 p^{2} T^{10} + 4 p^{3} T^{11} + 8 p^{4} T^{12} - 7 p^{5} T^{13} - p^{7} T^{15} + p^{8} T^{16} \) |
| 7 | \( ( 1 - 6 T + 3 p T^{2} - 44 T^{3} + 90 T^{4} - 44 p T^{5} + 3 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 11 | \( ( 1 - 5 T + 24 T^{2} - 42 T^{3} + 169 T^{4} - 42 p T^{5} + 24 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 13 | \( 1 - 9 T + 28 T^{2} - 63 T^{3} + 160 T^{4} + 486 T^{5} - 5114 T^{6} + 22158 T^{7} - 86609 T^{8} + 22158 p T^{9} - 5114 p^{2} T^{10} + 486 p^{3} T^{11} + 160 p^{4} T^{12} - 63 p^{5} T^{13} + 28 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 17 | \( 1 - 5 T - 15 T^{2} + 160 T^{3} - 307 T^{4} - 190 T^{5} - 60 T^{6} - 18825 T^{7} + 162978 T^{8} - 18825 p T^{9} - 60 p^{2} T^{10} - 190 p^{3} T^{11} - 307 p^{4} T^{12} + 160 p^{5} T^{13} - 15 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) | |
| 23 | \( ( 1 - 3 T - T^{2} + 108 T^{3} - 636 T^{4} + 108 p T^{5} - p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 29 | \( 1 - 17 T + 94 T^{2} - 357 T^{3} + 3704 T^{4} - 19244 T^{5} - 3426 T^{6} - 1700 p T^{7} + 2210431 T^{8} - 1700 p^{2} T^{9} - 3426 p^{2} T^{10} - 19244 p^{3} T^{11} + 3704 p^{4} T^{12} - 357 p^{5} T^{13} + 94 p^{6} T^{14} - 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 31 | \( ( 1 - 11 T + 83 T^{2} - 663 T^{3} + 4342 T^{4} - 663 p T^{5} + 83 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 37 | \( ( 1 + 4 T + 38 T^{2} + 216 T^{3} + 3170 T^{4} + 216 p T^{5} + 38 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 41 | \( 1 - 7 T - 15 T^{2} - 808 T^{3} + 5909 T^{4} + 11158 T^{5} + 334620 T^{6} - 2279715 T^{7} - 3979470 T^{8} - 2279715 p T^{9} + 334620 p^{2} T^{10} + 11158 p^{3} T^{11} + 5909 p^{4} T^{12} - 808 p^{5} T^{13} - 15 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) | |
| 43 | \( 1 + 13 T - 4 T^{2} - 505 T^{3} + 1020 T^{4} + 11224 T^{5} - 138244 T^{6} - 3186 p T^{7} + 180427 p T^{8} - 3186 p^{2} T^{9} - 138244 p^{2} T^{10} + 11224 p^{3} T^{11} + 1020 p^{4} T^{12} - 505 p^{5} T^{13} - 4 p^{6} T^{14} + 13 p^{7} T^{15} + p^{8} T^{16} \) | |
| 47 | \( 1 - 14 T - 45 T^{2} + 562 T^{3} + 12527 T^{4} - 62728 T^{5} - 695664 T^{6} + 248448 T^{7} + 49557882 T^{8} + 248448 p T^{9} - 695664 p^{2} T^{10} - 62728 p^{3} T^{11} + 12527 p^{4} T^{12} + 562 p^{5} T^{13} - 45 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 53 | \( 1 + 14 T - 69 T^{2} - 730 T^{3} + 319 p T^{4} + 83296 T^{5} - 1123140 T^{6} - 485772 T^{7} + 86530722 T^{8} - 485772 p T^{9} - 1123140 p^{2} T^{10} + 83296 p^{3} T^{11} + 319 p^{5} T^{12} - 730 p^{5} T^{13} - 69 p^{6} T^{14} + 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 59 | \( 1 - 14 T - 78 T^{2} + 796 T^{3} + 17885 T^{4} - 77752 T^{5} - 1427730 T^{6} + 1292118 T^{7} + 103988436 T^{8} + 1292118 p T^{9} - 1427730 p^{2} T^{10} - 77752 p^{3} T^{11} + 17885 p^{4} T^{12} + 796 p^{5} T^{13} - 78 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 61 | \( 1 + 9 T - 126 T^{2} - 1175 T^{3} + 11382 T^{4} + 86764 T^{5} - 669848 T^{6} - 2353650 T^{7} + 39432547 T^{8} - 2353650 p T^{9} - 669848 p^{2} T^{10} + 86764 p^{3} T^{11} + 11382 p^{4} T^{12} - 1175 p^{5} T^{13} - 126 p^{6} T^{14} + 9 p^{7} T^{15} + p^{8} T^{16} \) | |
| 67 | \( 1 - 6 T - 225 T^{2} + 818 T^{3} + 34977 T^{4} - 79072 T^{5} - 3498974 T^{6} + 1872192 T^{7} + 276094822 T^{8} + 1872192 p T^{9} - 3498974 p^{2} T^{10} - 79072 p^{3} T^{11} + 34977 p^{4} T^{12} + 818 p^{5} T^{13} - 225 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) | |
| 71 | \( 1 - 14 T - 45 T^{2} + 622 T^{3} + 7661 T^{4} + 5168 T^{5} - 974214 T^{6} + 1505364 T^{7} + 42583530 T^{8} + 1505364 p T^{9} - 974214 p^{2} T^{10} + 5168 p^{3} T^{11} + 7661 p^{4} T^{12} + 622 p^{5} T^{13} - 45 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 73 | \( 1 + 11 T - 118 T^{2} - 317 T^{3} + 19274 T^{4} - 16592 T^{5} - 1638744 T^{6} - 1228122 T^{7} + 71418583 T^{8} - 1228122 p T^{9} - 1638744 p^{2} T^{10} - 16592 p^{3} T^{11} + 19274 p^{4} T^{12} - 317 p^{5} T^{13} - 118 p^{6} T^{14} + 11 p^{7} T^{15} + p^{8} T^{16} \) | |
| 79 | \( 1 + 17 T + 2 T^{2} - 1139 T^{3} + 1002 T^{4} + 74018 T^{5} - 76090 T^{6} - 6700686 T^{7} - 70425071 T^{8} - 6700686 p T^{9} - 76090 p^{2} T^{10} + 74018 p^{3} T^{11} + 1002 p^{4} T^{12} - 1139 p^{5} T^{13} + 2 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} \) | |
| 83 | \( ( 1 + 23 T + 463 T^{2} + 5583 T^{3} + 61208 T^{4} + 5583 p T^{5} + 463 p^{2} T^{6} + 23 p^{3} T^{7} + p^{4} T^{8} )^{2} \) | |
| 89 | \( 1 - 14 T - 116 T^{2} + 888 T^{3} + 23783 T^{4} - 24296 T^{5} - 2897328 T^{6} + 4851182 T^{7} + 179834968 T^{8} + 4851182 p T^{9} - 2897328 p^{2} T^{10} - 24296 p^{3} T^{11} + 23783 p^{4} T^{12} + 888 p^{5} T^{13} - 116 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) | |
| 97 | \( 1 + 17 T - 109 T^{2} - 3026 T^{3} + 16787 T^{4} + 438430 T^{5} + 149646 T^{6} - 13000257 T^{7} + 32211550 T^{8} - 13000257 p T^{9} + 149646 p^{2} T^{10} + 438430 p^{3} T^{11} + 16787 p^{4} T^{12} - 3026 p^{5} T^{13} - 109 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} \) | |
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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
−4.34265134707071256253182643195, −4.24468052797935756260671612964, −4.22387095710831998726944779110, −4.02602054700820808097481014772, −3.88393314550382018891696871712, −3.63674442117526412307552937361, −3.56381278594429813929473774706, −3.25865818356323001423962852619, −3.18618113499131837528109463500, −2.98338195675473807541413315517, −2.85505980203382744074107877624, −2.75765289535690164328514817181, −2.69322067094644273234271379910, −2.37259887063712943202966284659, −2.02599698106558844603331407798, −1.90932111083620317694473839168, −1.75037563906821826132518009970, −1.53643610805728103903070075573, −1.44430552970843644137409266123, −1.42133825423053151649002018998, −1.14077067488481086075913592316, −1.10986719782811446732585599109, −0.968694144587783523765367278157, −0.74040637654630414715436523772, −0.67628866366823084717572769246, 0.67628866366823084717572769246, 0.74040637654630414715436523772, 0.968694144587783523765367278157, 1.10986719782811446732585599109, 1.14077067488481086075913592316, 1.42133825423053151649002018998, 1.44430552970843644137409266123, 1.53643610805728103903070075573, 1.75037563906821826132518009970, 1.90932111083620317694473839168, 2.02599698106558844603331407798, 2.37259887063712943202966284659, 2.69322067094644273234271379910, 2.75765289535690164328514817181, 2.85505980203382744074107877624, 2.98338195675473807541413315517, 3.18618113499131837528109463500, 3.25865818356323001423962852619, 3.56381278594429813929473774706, 3.63674442117526412307552937361, 3.88393314550382018891696871712, 4.02602054700820808097481014772, 4.22387095710831998726944779110, 4.24468052797935756260671612964, 4.34265134707071256253182643195
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.