Properties

Label 4-2394e2-1.1-c1e2-0-15
Degree $4$
Conductor $5731236$
Sign $1$
Analytic cond. $365.428$
Root an. cond. $4.37220$
Motivic weight $1$
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 + 3·4-s − 2·7-s + 4·8-s + 4·11-s − 4·14-s + 5·16-s − 8·17-s + 2·19-s + 8·22-s + 4·23-s + 2·25-s − 6·28-s + 12·29-s + 6·32-s − 16·34-s + 20·37-s + 4·38-s + 4·41-s + 12·44-s + 8·46-s − 8·47-s + 3·49-s + 4·50-s − 4·53-s − 8·56-s + 24·58-s + ⋯
L(s)  = 1  + 1.41·2-s + 3/2·4-s − 0.755·7-s + 1.41·8-s + 1.20·11-s − 1.06·14-s + 5/4·16-s − 1.94·17-s + 0.458·19-s + 1.70·22-s + 0.834·23-s + 2/5·25-s − 1.13·28-s + 2.22·29-s + 1.06·32-s − 2.74·34-s + 3.28·37-s + 0.648·38-s + 0.624·41-s + 1.80·44-s + 1.17·46-s − 1.16·47-s + 3/7·49-s + 0.565·50-s − 0.549·53-s − 1.06·56-s + 3.15·58-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5731236 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5731236 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5731236\)    =    \(2^{2} \cdot 3^{4} \cdot 7^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(365.428\)
Root analytic conductor: \(4.37220\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{2394} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5731236,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.426015359\)
\(L(\frac12)\) \(\approx\) \(7.426015359\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
3 \( 1 \)
7$C_1$ \( ( 1 + T )^{2} \)
19$C_1$ \( ( 1 - T )^{2} \)
good5$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - 4 T + 14 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
61$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 12 T + 158 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 8 T + 110 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
79$D_{4}$ \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
97$D_{4}$ \( 1 - 16 T + 246 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
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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

−9.285055372263357549392046008434, −8.775042265480180355137739906589, −8.297782014101703106788659932133, −8.000370395302103329151571202217, −7.48811881540264659913874606409, −6.78261303430059598583977675594, −6.68304401168267273880138709452, −6.61077543430544603923983161339, −5.98159371006717615689971220033, −5.83723660437817179087812207865, −4.87709161993679250703606891524, −4.82765091075040148017849262651, −4.40160320314858647422857199763, −4.00282102307325698169408895488, −3.48726672630995003318327120659, −3.03993034257982394543822731262, −2.48624011177232257523212804641, −2.31142045400819991841852134076, −1.26988506724232690547654711275, −0.793184999774424812735472750690, 0.793184999774424812735472750690, 1.26988506724232690547654711275, 2.31142045400819991841852134076, 2.48624011177232257523212804641, 3.03993034257982394543822731262, 3.48726672630995003318327120659, 4.00282102307325698169408895488, 4.40160320314858647422857199763, 4.82765091075040148017849262651, 4.87709161993679250703606891524, 5.83723660437817179087812207865, 5.98159371006717615689971220033, 6.61077543430544603923983161339, 6.68304401168267273880138709452, 6.78261303430059598583977675594, 7.48811881540264659913874606409, 8.000370395302103329151571202217, 8.297782014101703106788659932133, 8.775042265480180355137739906589, 9.285055372263357549392046008434

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