Dirichlet series
L(s) = 1 | + 1.21e5·2-s + 6.13e9·4-s + 1.81e11·5-s − 6.71e13·7-s + 7.37e14·8-s + 2.20e16·10-s − 1.33e17·11-s − 2.98e18·13-s − 8.17e18·14-s + 8.99e19·16-s + 7.93e19·17-s − 1.36e21·19-s + 1.11e21·20-s − 1.62e22·22-s − 2.61e22·23-s − 1.95e23·25-s − 3.62e23·26-s − 4.12e23·28-s + 1.67e24·29-s − 6.23e24·31-s + 1.99e24·32-s + 9.65e24·34-s − 1.21e25·35-s − 1.04e26·37-s − 1.65e26·38-s + 1.33e26·40-s − 2.77e26·41-s + ⋯ |
L(s) = 1 | + 1.31·2-s + 0.714·4-s + 0.530·5-s − 0.763·7-s + 0.926·8-s + 0.696·10-s − 0.878·11-s − 1.24·13-s − 1.00·14-s + 1.21·16-s + 0.395·17-s − 1.08·19-s + 0.379·20-s − 1.15·22-s − 0.889·23-s − 1.68·25-s − 1.63·26-s − 0.545·28-s + 1.24·29-s − 1.54·31-s + 0.290·32-s + 0.519·34-s − 0.405·35-s − 1.39·37-s − 1.42·38-s + 0.491·40-s − 0.679·41-s + ⋯ |
Functional equation
Invariants
Degree: | \(4\) |
Conductor: | \(81\) = \(3^{4}\) |
Sign: | $1$ |
Analytic conductor: | \(3854.49\) |
Root analytic conductor: | \(7.87937\) |
Motivic weight: | \(33\) |
Rational: | yes |
Arithmetic: | yes |
Character: | Trivial |
Primitive: | no |
Self-dual: | yes |
Analytic rank: | \(0\) |
Selberg data: | \((4,\ 81,\ (\ :33/2, 33/2),\ 1)\) |
Particular Values
\(L(17)\) | \(\approx\) | \(0.02121514055\) |
\(L(\frac12)\) | \(\approx\) | \(0.02121514055\) |
\(L(\frac{35}{2})\) | not available | |
\(L(1)\) | not available |
Euler product
$p$ | $\Gal(F_p)$ | $F_p(T)$ | |
---|---|---|---|
bad | 3 | \( 1 \) | |
good | 2 | $D_{4}$ | \( 1 - 7605 p^{4} T + 2115925 p^{12} T^{2} - 7605 p^{37} T^{3} + p^{66} T^{4} \) |
5 | $D_{4}$ | \( 1 - 1448492292 p^{3} T + 585507855025144558 p^{8} T^{2} - 1448492292 p^{36} T^{3} + p^{66} T^{4} \) | |
7 | $D_{4}$ | \( 1 + 9593297152400 p T + \)\(36\!\cdots\!50\)\( p^{4} T^{2} + 9593297152400 p^{34} T^{3} + p^{66} T^{4} \) | |
11 | $D_{4}$ | \( 1 + 12170165040174024 p T + \)\(23\!\cdots\!66\)\( p^{2} T^{2} + 12170165040174024 p^{34} T^{3} + p^{66} T^{4} \) | |
13 | $D_{4}$ | \( 1 + 229354652173803380 p T + \)\(61\!\cdots\!50\)\( p^{3} T^{2} + 229354652173803380 p^{34} T^{3} + p^{66} T^{4} \) | |
17 | $D_{4}$ | \( 1 - 79361149261175525340 T - \)\(26\!\cdots\!50\)\( p T^{2} - 79361149261175525340 p^{33} T^{3} + p^{66} T^{4} \) | |
19 | $D_{4}$ | \( 1 + \)\(13\!\cdots\!00\)\( T + \)\(17\!\cdots\!22\)\( p T^{2} + \)\(13\!\cdots\!00\)\( p^{33} T^{3} + p^{66} T^{4} \) | |
23 | $D_{4}$ | \( 1 + \)\(26\!\cdots\!40\)\( T + \)\(68\!\cdots\!50\)\( p T^{2} + \)\(26\!\cdots\!40\)\( p^{33} T^{3} + p^{66} T^{4} \) | |
29 | $D_{4}$ | \( 1 - \)\(57\!\cdots\!00\)\( p T + \)\(46\!\cdots\!58\)\( p^{2} T^{2} - \)\(57\!\cdots\!00\)\( p^{34} T^{3} + p^{66} T^{4} \) | |
31 | $D_{4}$ | \( 1 + \)\(20\!\cdots\!36\)\( p T + \)\(30\!\cdots\!86\)\( p^{2} T^{2} + \)\(20\!\cdots\!36\)\( p^{34} T^{3} + p^{66} T^{4} \) | |
37 | $D_{4}$ | \( 1 + \)\(10\!\cdots\!20\)\( T + \)\(13\!\cdots\!50\)\( T^{2} + \)\(10\!\cdots\!20\)\( p^{33} T^{3} + p^{66} T^{4} \) | |
41 | $D_{4}$ | \( 1 + \)\(27\!\cdots\!44\)\( T + \)\(22\!\cdots\!26\)\( T^{2} + \)\(27\!\cdots\!44\)\( p^{33} T^{3} + p^{66} T^{4} \) | |
43 | $D_{4}$ | \( 1 - \)\(15\!\cdots\!00\)\( T + \)\(12\!\cdots\!50\)\( T^{2} - \)\(15\!\cdots\!00\)\( p^{33} T^{3} + p^{66} T^{4} \) | |
47 | $D_{4}$ | \( 1 + \)\(11\!\cdots\!20\)\( p T + \)\(37\!\cdots\!50\)\( T^{2} + \)\(11\!\cdots\!20\)\( p^{34} T^{3} + p^{66} T^{4} \) | |
53 | $D_{4}$ | \( 1 - \)\(26\!\cdots\!20\)\( T + \)\(12\!\cdots\!50\)\( T^{2} - \)\(26\!\cdots\!20\)\( p^{33} T^{3} + p^{66} T^{4} \) | |
59 | $D_{4}$ | \( 1 - \)\(30\!\cdots\!00\)\( T + \)\(77\!\cdots\!58\)\( T^{2} - \)\(30\!\cdots\!00\)\( p^{33} T^{3} + p^{66} T^{4} \) | |
61 | $D_{4}$ | \( 1 + \)\(57\!\cdots\!36\)\( T + \)\(16\!\cdots\!86\)\( T^{2} + \)\(57\!\cdots\!36\)\( p^{33} T^{3} + p^{66} T^{4} \) | |
67 | $D_{4}$ | \( 1 - \)\(15\!\cdots\!60\)\( T + \)\(41\!\cdots\!50\)\( T^{2} - \)\(15\!\cdots\!60\)\( p^{33} T^{3} + p^{66} T^{4} \) | |
71 | $D_{4}$ | \( 1 - \)\(26\!\cdots\!76\)\( T + \)\(22\!\cdots\!66\)\( T^{2} - \)\(26\!\cdots\!76\)\( p^{33} T^{3} + p^{66} T^{4} \) | |
73 | $D_{4}$ | \( 1 - \)\(94\!\cdots\!40\)\( T + \)\(19\!\cdots\!50\)\( T^{2} - \)\(94\!\cdots\!40\)\( p^{33} T^{3} + p^{66} T^{4} \) | |
79 | $D_{4}$ | \( 1 + \)\(85\!\cdots\!00\)\( T + \)\(85\!\cdots\!78\)\( T^{2} + \)\(85\!\cdots\!00\)\( p^{33} T^{3} + p^{66} T^{4} \) | |
83 | $D_{4}$ | \( 1 + \)\(29\!\cdots\!20\)\( T + \)\(37\!\cdots\!50\)\( T^{2} + \)\(29\!\cdots\!20\)\( p^{33} T^{3} + p^{66} T^{4} \) | |
89 | $D_{4}$ | \( 1 + \)\(13\!\cdots\!00\)\( T + \)\(47\!\cdots\!38\)\( T^{2} + \)\(13\!\cdots\!00\)\( p^{33} T^{3} + p^{66} T^{4} \) | |
97 | $D_{4}$ | \( 1 + \)\(36\!\cdots\!60\)\( T + \)\(42\!\cdots\!50\)\( T^{2} + \)\(36\!\cdots\!60\)\( p^{33} T^{3} + p^{66} T^{4} \) | |
show more | |||
show less |
Imaginary part of the first few zeros on the critical line
−14.08015330685980784913447413397, −13.46912948139148517203353618901, −12.76814587770784757622545255447, −12.54422226576142171013309887400, −11.65882201042686300120446471056, −10.62469884087428772281734725786, −10.07289973263252693515985927038, −9.622016537176720860979209534186, −8.411574983212849430759663194070, −7.66309016926825645684712359703, −6.93953776391012555694629037343, −6.18307145153677745740702837569, −5.26228145783937502490386112774, −5.19962423923673568072910494536, −4.00795900289828917689815351764, −3.77175510616389440508152230795, −2.62532772055263355972238419670, −2.21738605633794544192925859210, −1.46123154365470444616523474221, −0.02658994385999218551493683890, 0.02658994385999218551493683890, 1.46123154365470444616523474221, 2.21738605633794544192925859210, 2.62532772055263355972238419670, 3.77175510616389440508152230795, 4.00795900289828917689815351764, 5.19962423923673568072910494536, 5.26228145783937502490386112774, 6.18307145153677745740702837569, 6.93953776391012555694629037343, 7.66309016926825645684712359703, 8.411574983212849430759663194070, 9.622016537176720860979209534186, 10.07289973263252693515985927038, 10.62469884087428772281734725786, 11.65882201042686300120446471056, 12.54422226576142171013309887400, 12.76814587770784757622545255447, 13.46912948139148517203353618901, 14.08015330685980784913447413397