Dirichlet series
L(s) = 1 | − 1.73e7·3-s − 1.93e10·5-s − 3.02e13·7-s − 5.17e14·9-s + 7.78e15·11-s + 7.47e16·13-s + 3.36e17·15-s + 1.72e19·17-s + 1.23e19·19-s + 5.25e20·21-s − 1.89e21·23-s − 3.72e21·25-s + 1.24e22·27-s + 1.28e23·29-s − 1.25e23·31-s − 1.35e23·33-s + 5.86e23·35-s − 8.33e23·37-s − 1.29e24·39-s + 8.72e24·41-s + 1.83e25·43-s + 1.00e25·45-s − 9.54e25·47-s + 3.84e26·49-s − 2.99e26·51-s + 1.94e26·53-s − 1.50e26·55-s + ⋯ |
L(s) = 1 | − 0.698·3-s − 0.284·5-s − 2.40·7-s − 0.837·9-s + 0.561·11-s + 0.404·13-s + 0.198·15-s + 1.45·17-s + 0.186·19-s + 1.68·21-s − 1.48·23-s − 0.800·25-s + 0.813·27-s + 2.76·29-s − 0.962·31-s − 0.392·33-s + 0.684·35-s − 0.411·37-s − 0.282·39-s + 0.876·41-s + 0.883·43-s + 0.238·45-s − 1.15·47-s + 2.43·49-s − 1.01·51-s + 0.365·53-s − 0.159·55-s + ⋯ |
Functional equation
Invariants
Degree: | \(4\) |
Conductor: | \(256\) = \(2^{8}\) |
Sign: | $1$ |
Analytic conductor: | \(9487.42\) |
Root analytic conductor: | \(9.86931\) |
Motivic weight: | \(31\) |
Rational: | yes |
Arithmetic: | yes |
Character: | Trivial |
Primitive: | no |
Self-dual: | yes |
Analytic rank: | \(2\) |
Selberg data: | \((4,\ 256,\ (\ :31/2, 31/2),\ 1)\) |
Particular Values
\(L(16)\) | \(=\) | \(0\) |
\(L(\frac12)\) | \(=\) | \(0\) |
\(L(\frac{33}{2})\) | not available | |
\(L(1)\) | not available |
Euler product
$p$ | $\Gal(F_p)$ | $F_p(T)$ | |
---|---|---|---|
bad | 2 | \( 1 \) | |
good | 3 | $D_{4}$ | \( 1 + 214360 p^{4} T + 13870620790 p^{10} T^{2} + 214360 p^{35} T^{3} + p^{62} T^{4} \) |
5 | $D_{4}$ | \( 1 + 3878243604 p T + 1313821413051106982 p^{5} T^{2} + 3878243604 p^{32} T^{3} + p^{62} T^{4} \) | |
7 | $D_{4}$ | \( 1 + 617500562800 p^{2} T + \)\(31\!\cdots\!50\)\( p^{5} T^{2} + 617500562800 p^{33} T^{3} + p^{62} T^{4} \) | |
11 | $D_{4}$ | \( 1 - 7782353745118776 T + \)\(32\!\cdots\!46\)\( p^{2} T^{2} - 7782353745118776 p^{31} T^{3} + p^{62} T^{4} \) | |
13 | $D_{4}$ | \( 1 - 5746842542327740 p T + \)\(55\!\cdots\!10\)\( p^{3} T^{2} - 5746842542327740 p^{32} T^{3} + p^{62} T^{4} \) | |
17 | $D_{4}$ | \( 1 - 17224607828987089380 T + \)\(20\!\cdots\!70\)\( p T^{2} - 17224607828987089380 p^{31} T^{3} + p^{62} T^{4} \) | |
19 | $D_{4}$ | \( 1 - 651082280422219160 p T + \)\(24\!\cdots\!58\)\( p^{2} T^{2} - 651082280422219160 p^{32} T^{3} + p^{62} T^{4} \) | |
23 | $D_{4}$ | \( 1 + 82493253999561357360 p T + \)\(61\!\cdots\!70\)\( p^{2} T^{2} + 82493253999561357360 p^{32} T^{3} + p^{62} T^{4} \) | |
29 | $D_{4}$ | \( 1 - \)\(44\!\cdots\!60\)\( p T + \)\(98\!\cdots\!38\)\( p^{2} T^{2} - \)\(44\!\cdots\!60\)\( p^{32} T^{3} + p^{62} T^{4} \) | |
31 | $D_{4}$ | \( 1 + \)\(12\!\cdots\!64\)\( T + \)\(22\!\cdots\!86\)\( T^{2} + \)\(12\!\cdots\!64\)\( p^{31} T^{3} + p^{62} T^{4} \) | |
37 | $D_{4}$ | \( 1 + \)\(83\!\cdots\!60\)\( T + \)\(56\!\cdots\!70\)\( T^{2} + \)\(83\!\cdots\!60\)\( p^{31} T^{3} + p^{62} T^{4} \) | |
41 | $D_{4}$ | \( 1 - \)\(87\!\cdots\!84\)\( T + \)\(12\!\cdots\!46\)\( T^{2} - \)\(87\!\cdots\!84\)\( p^{31} T^{3} + p^{62} T^{4} \) | |
43 | $D_{4}$ | \( 1 - \)\(18\!\cdots\!00\)\( T + \)\(64\!\cdots\!50\)\( T^{2} - \)\(18\!\cdots\!00\)\( p^{31} T^{3} + p^{62} T^{4} \) | |
47 | $D_{4}$ | \( 1 + \)\(95\!\cdots\!20\)\( T + \)\(15\!\cdots\!10\)\( T^{2} + \)\(95\!\cdots\!20\)\( p^{31} T^{3} + p^{62} T^{4} \) | |
53 | $D_{4}$ | \( 1 - \)\(19\!\cdots\!60\)\( T + \)\(57\!\cdots\!10\)\( T^{2} - \)\(19\!\cdots\!60\)\( p^{31} T^{3} + p^{62} T^{4} \) | |
59 | $D_{4}$ | \( 1 - \)\(19\!\cdots\!20\)\( T + \)\(10\!\cdots\!18\)\( T^{2} - \)\(19\!\cdots\!20\)\( p^{31} T^{3} + p^{62} T^{4} \) | |
61 | $D_{4}$ | \( 1 + \)\(12\!\cdots\!76\)\( T + \)\(80\!\cdots\!66\)\( T^{2} + \)\(12\!\cdots\!76\)\( p^{31} T^{3} + p^{62} T^{4} \) | |
67 | $D_{4}$ | \( 1 - \)\(96\!\cdots\!20\)\( T + \)\(81\!\cdots\!90\)\( T^{2} - \)\(96\!\cdots\!20\)\( p^{31} T^{3} + p^{62} T^{4} \) | |
71 | $D_{4}$ | \( 1 + \)\(55\!\cdots\!44\)\( T + \)\(32\!\cdots\!26\)\( T^{2} + \)\(55\!\cdots\!44\)\( p^{31} T^{3} + p^{62} T^{4} \) | |
73 | $D_{4}$ | \( 1 - \)\(62\!\cdots\!80\)\( T + \)\(12\!\cdots\!30\)\( T^{2} - \)\(62\!\cdots\!80\)\( p^{31} T^{3} + p^{62} T^{4} \) | |
79 | $D_{4}$ | \( 1 - \)\(11\!\cdots\!60\)\( T + \)\(80\!\cdots\!58\)\( T^{2} - \)\(11\!\cdots\!60\)\( p^{31} T^{3} + p^{62} T^{4} \) | |
83 | $D_{4}$ | \( 1 - \)\(26\!\cdots\!60\)\( T + \)\(53\!\cdots\!90\)\( T^{2} - \)\(26\!\cdots\!60\)\( p^{31} T^{3} + p^{62} T^{4} \) | |
89 | $D_{4}$ | \( 1 + \)\(21\!\cdots\!80\)\( T + \)\(47\!\cdots\!78\)\( T^{2} + \)\(21\!\cdots\!80\)\( p^{31} T^{3} + p^{62} T^{4} \) | |
97 | $D_{4}$ | \( 1 + \)\(90\!\cdots\!80\)\( T + \)\(97\!\cdots\!10\)\( T^{2} + \)\(90\!\cdots\!80\)\( p^{31} T^{3} + p^{62} T^{4} \) | |
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Imaginary part of the first few zeros on the critical line
−12.06131541669637342874102915058, −11.99535318983584107823397307361, −10.88178964821442113877632674814, −10.33232037578400838041149354325, −9.596513020634365794345730537882, −9.382570253852432803683094582159, −8.336628187745705255774803512173, −7.80329609310725445955687260468, −6.70385038716037416483855083404, −6.44565962062403862600909705905, −5.82787808634420954040231082588, −5.47060796249019633993679258641, −4.23400058602786436934471680550, −3.73165511717929586206272876990, −2.93981796131718331783909151817, −2.85503956122382287994927217842, −1.51836086979972854635709210098, −0.817508287824355137011606373964, 0, 0, 0.817508287824355137011606373964, 1.51836086979972854635709210098, 2.85503956122382287994927217842, 2.93981796131718331783909151817, 3.73165511717929586206272876990, 4.23400058602786436934471680550, 5.47060796249019633993679258641, 5.82787808634420954040231082588, 6.44565962062403862600909705905, 6.70385038716037416483855083404, 7.80329609310725445955687260468, 8.336628187745705255774803512173, 9.382570253852432803683094582159, 9.596513020634365794345730537882, 10.33232037578400838041149354325, 10.88178964821442113877632674814, 11.99535318983584107823397307361, 12.06131541669637342874102915058