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Properties

Label 4-39e4-1.1-c0e2-0-0
Degree $4$
Conductor $2313441$
Sign $1$
Analytic cond. $0.576199$
Root an. cond. $0.871250$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s + 2·5-s − 2·8-s − 4·10-s − 2·11-s + 3·16-s + 4·20-s + 4·22-s + 2·25-s − 4·32-s − 4·40-s + 2·41-s − 4·44-s − 2·47-s − 4·50-s − 4·55-s + 2·59-s + 4·64-s + 2·71-s + 6·80-s − 4·82-s − 2·83-s + 4·88-s + 2·89-s + 4·94-s + 4·100-s + ⋯
L(s)  = 1  − 2·2-s + 2·4-s + 2·5-s − 2·8-s − 4·10-s − 2·11-s + 3·16-s + 4·20-s + 4·22-s + 2·25-s − 4·32-s − 4·40-s + 2·41-s − 4·44-s − 2·47-s − 4·50-s − 4·55-s + 2·59-s + 4·64-s + 2·71-s + 6·80-s − 4·82-s − 2·83-s + 4·88-s + 2·89-s + 4·94-s + 4·100-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(2313441\)    =    \(3^{4} \cdot 13^{4}\)
Sign: $1$
Analytic conductor: \(0.576199\)
Root analytic conductor: \(0.871250\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 2313441,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4424510520\)
\(L(\frac12)\) \(\approx\) \(0.4424510520\)
\(L(1)\) 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)$
bad3 \( 1 \)
13 \( 1 \)
good2$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
5$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
7$C_2^2$ \( 1 + T^{4} \)
11$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
17$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
19$C_2^2$ \( 1 + T^{4} \)
23$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
29$C_2$ \( ( 1 + T^{2} )^{2} \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_2^2$ \( 1 + T^{4} \)
41$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
43$C_2$ \( ( 1 + T^{2} )^{2} \)
47$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
53$C_2$ \( ( 1 + T^{2} )^{2} \)
59$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
61$C_2$ \( ( 1 + T^{2} )^{2} \)
67$C_2^2$ \( 1 + T^{4} \)
71$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
73$C_2^2$ \( 1 + T^{4} \)
79$C_2$ \( ( 1 + T^{2} )^{2} \)
83$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
89$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
97$C_2^2$ \( 1 + T^{4} \)
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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

−9.784916867215331326238519325768, −9.385502418663857334519916812848, −9.341202702754915649874651990744, −8.705733363770225578883724166419, −8.252786851167246921159116043252, −8.192329102445047568091107281418, −7.54807602219051054544317013923, −7.34551415863903369728697565142, −6.51889451029998174486108124524, −6.43545905638323584997957937306, −5.71928271138672447523973878824, −5.53749890009658633194508693005, −5.24258287275329525329409451109, −4.61359621105249802277571192052, −3.65644961633868683754279358263, −3.04860075508357261789170127706, −2.50821271977971702998372531733, −2.21690085741720950086207860267, −1.57872489189144797992384118298, −0.75235534236804059763170543133, 0.75235534236804059763170543133, 1.57872489189144797992384118298, 2.21690085741720950086207860267, 2.50821271977971702998372531733, 3.04860075508357261789170127706, 3.65644961633868683754279358263, 4.61359621105249802277571192052, 5.24258287275329525329409451109, 5.53749890009658633194508693005, 5.71928271138672447523973878824, 6.43545905638323584997957937306, 6.51889451029998174486108124524, 7.34551415863903369728697565142, 7.54807602219051054544317013923, 8.192329102445047568091107281418, 8.252786851167246921159116043252, 8.705733363770225578883724166419, 9.341202702754915649874651990744, 9.385502418663857334519916812848, 9.784916867215331326238519325768

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