Properties

Label 4-280e2-1.1-c0e2-0-1
Degree $4$
Conductor $78400$
Sign $1$
Analytic cond. $0.0195267$
Root an. cond. $0.373815$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 5-s − 2·7-s − 8-s − 9-s + 10-s + 11-s + 2·13-s − 2·14-s − 16-s − 18-s + 19-s + 22-s − 23-s + 2·26-s − 2·35-s − 37-s + 38-s − 40-s − 2·41-s − 45-s − 46-s − 47-s + 3·49-s − 53-s + 55-s + 2·56-s + ⋯
L(s)  = 1  + 2-s + 5-s − 2·7-s − 8-s − 9-s + 10-s + 11-s + 2·13-s − 2·14-s − 16-s − 18-s + 19-s + 22-s − 23-s + 2·26-s − 2·35-s − 37-s + 38-s − 40-s − 2·41-s − 45-s − 46-s − 47-s + 3·49-s − 53-s + 55-s + 2·56-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(78400\)    =    \(2^{6} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(0.0195267\)
Root analytic conductor: \(0.373815\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{280} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 78400,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8106410075\)
\(L(\frac12)\) \(\approx\) \(0.8106410075\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$\Gal(F_p)$$F_p(T)$
bad2$C_2$ \( 1 - T + T^{2} \)
5$C_2$ \( 1 - T + T^{2} \)
7$C_1$ \( ( 1 + T )^{2} \)
good3$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
11$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
13$C_2$ \( ( 1 - T + T^{2} )^{2} \)
17$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T + T^{2} ) \)
23$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
29$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
31$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
41$C_2$ \( ( 1 + T + T^{2} )^{2} \)
43$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
47$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
53$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 - T + T^{2} ) \)
59$C_2$ \( ( 1 + T + T^{2} )^{2} \)
61$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
67$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
71$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
73$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
79$C_2$ \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \)
83$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
89$C_2$ \( ( 1 + T + T^{2} )^{2} \)
97$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.43661702903630611698648175216, −11.95801266419308080269532549316, −11.58228246400703646156965992564, −11.05087231705270468550520864070, −10.34981008285597145419030574862, −9.899670395063341170480745356369, −9.367759933787970351658553263718, −9.259868635247810357625713530602, −8.494984857385028792438112678468, −8.340390900610433125282624132098, −7.08164090917019720194429738367, −6.60667523442364763206691892267, −6.14344346913775295094976015169, −5.88650311431721599084779389603, −5.57308134796154957948202176018, −4.61786745847556869163907995676, −3.76275309843798213587506784737, −3.27215264224217588357928356523, −3.13096812951106527186115780787, −1.75406181552643421361357220844, 1.75406181552643421361357220844, 3.13096812951106527186115780787, 3.27215264224217588357928356523, 3.76275309843798213587506784737, 4.61786745847556869163907995676, 5.57308134796154957948202176018, 5.88650311431721599084779389603, 6.14344346913775295094976015169, 6.60667523442364763206691892267, 7.08164090917019720194429738367, 8.340390900610433125282624132098, 8.494984857385028792438112678468, 9.259868635247810357625713530602, 9.367759933787970351658553263718, 9.899670395063341170480745356369, 10.34981008285597145419030574862, 11.05087231705270468550520864070, 11.58228246400703646156965992564, 11.95801266419308080269532549316, 12.43661702903630611698648175216

Graph of the $Z$-function along the critical line