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Properties

Label 4-2349e2-1.1-c0e2-0-5
Degree $4$
Conductor $5517801$
Sign $1$
Analytic cond. $1.37429$
Root an. cond. $1.08272$
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  + 2·2-s + 2·4-s + 2·7-s + 2·8-s + 4·14-s + 3·16-s − 2·17-s + 2·19-s − 2·23-s + 25-s + 4·28-s + 4·32-s − 4·34-s − 2·37-s + 4·38-s − 2·43-s − 4·46-s + 49-s + 2·50-s − 2·53-s + 4·56-s − 2·59-s + 4·64-s − 4·68-s + 2·73-s − 4·74-s + 4·76-s + ⋯
L(s)  = 1  + 2·2-s + 2·4-s + 2·7-s + 2·8-s + 4·14-s + 3·16-s − 2·17-s + 2·19-s − 2·23-s + 25-s + 4·28-s + 4·32-s − 4·34-s − 2·37-s + 4·38-s − 2·43-s − 4·46-s + 49-s + 2·50-s − 2·53-s + 4·56-s − 2·59-s + 4·64-s − 4·68-s + 2·73-s − 4·74-s + 4·76-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5517801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5517801 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5517801\)    =    \(3^{8} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(1.37429\)
Root analytic conductor: \(1.08272\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5517801,\ (\ :0, 0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.128453349\)
\(L(\frac12)\) \(\approx\) \(5.128453349\)
\(L(1)\) 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 \( 1 \)
29$C_2$ \( 1 + T^{2} \)
good2$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
5$C_2^2$ \( 1 - T^{2} + T^{4} \)
7$C_2$ \( ( 1 - T + T^{2} )^{2} \)
11$C_2^2$ \( 1 + T^{4} \)
13$C_1$$\times$$C_1$ \( ( 1 - T )^{2}( 1 + T )^{2} \)
17$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
19$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
23$C_2$ \( ( 1 + T + T^{2} )^{2} \)
31$C_2^2$ \( 1 + T^{4} \)
37$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
41$C_2^2$ \( 1 + T^{4} \)
43$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
47$C_2^2$ \( 1 + T^{4} \)
53$C_2$ \( ( 1 + T + T^{2} )^{2} \)
59$C_2$ \( ( 1 + T + T^{2} )^{2} \)
61$C_2^2$ \( 1 + T^{4} \)
67$C_2^2$ \( 1 - T^{2} + T^{4} \)
71$C_2^2$ \( 1 - T^{2} + T^{4} \)
73$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
79$C_2^2$ \( 1 + T^{4} \)
83$C_2$ \( ( 1 - T + T^{2} )^{2} \)
89$C_1$$\times$$C_2$ \( ( 1 - T )^{2}( 1 + T^{2} ) \)
97$C_1$$\times$$C_2$ \( ( 1 + T )^{2}( 1 + T^{2} ) \)
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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

−9.345527143960472994072143793344, −8.922789044166219694883892714229, −8.323941367290746803611076620001, −8.081101954028420613815127041529, −7.76418342278291205998946309409, −7.55741616192090305467489733063, −6.83525011346803476766511595849, −6.39080318281557172706205654347, −6.37193587607691160501012073120, −5.40156028620408137858568708525, −5.26441515469981557978922361631, −4.88838097778145096990723349276, −4.82135509549536482722543992394, −4.20018951553101395807897265115, −3.92032339902164229411409218943, −3.26915596627658003732198862498, −3.03751656664131684398159270309, −2.05983775290663618795751287036, −1.78428359237476387138839733399, −1.35728935939910857885299747344, 1.35728935939910857885299747344, 1.78428359237476387138839733399, 2.05983775290663618795751287036, 3.03751656664131684398159270309, 3.26915596627658003732198862498, 3.92032339902164229411409218943, 4.20018951553101395807897265115, 4.82135509549536482722543992394, 4.88838097778145096990723349276, 5.26441515469981557978922361631, 5.40156028620408137858568708525, 6.37193587607691160501012073120, 6.39080318281557172706205654347, 6.83525011346803476766511595849, 7.55741616192090305467489733063, 7.76418342278291205998946309409, 8.081101954028420613815127041529, 8.323941367290746803611076620001, 8.922789044166219694883892714229, 9.345527143960472994072143793344

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