Properties

Label 4-280e2-1.1-c0e2-0-0
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

\(L(\frac{1}{2})\) \(\approx\) \(0.3243421534\)
\(L(\frac12)\) \(\approx\) \(0.3243421534\)
\(L(1)\) 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

−12.13830326851865339363536546666, −11.80071508954180853926845378276, −11.20615371565957059761013023842, −11.13932298076329041515698727402, −10.48062927226413804045761639682, −9.860082458716909779214524538077, −9.368086750882871791009580674703, −8.973991931372176462136622755714, −8.292953158098930586523783564286, −8.244697824609083528068906561575, −7.45877146856009408561287802246, −7.38314043813070187558221167778, −6.79736537383575133286069677998, −5.53551928462064910010219100853, −5.26808382279490391258148059126, −4.45199213217626417055744200907, −4.38038338526091381188467776575, −3.23446085663219780964219393327, −2.30247443887687932688881450063, −1.29490937665423030484310327694, 1.29490937665423030484310327694, 2.30247443887687932688881450063, 3.23446085663219780964219393327, 4.38038338526091381188467776575, 4.45199213217626417055744200907, 5.26808382279490391258148059126, 5.53551928462064910010219100853, 6.79736537383575133286069677998, 7.38314043813070187558221167778, 7.45877146856009408561287802246, 8.244697824609083528068906561575, 8.292953158098930586523783564286, 8.973991931372176462136622755714, 9.368086750882871791009580674703, 9.860082458716909779214524538077, 10.48062927226413804045761639682, 11.13932298076329041515698727402, 11.20615371565957059761013023842, 11.80071508954180853926845378276, 12.13830326851865339363536546666

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