Properties

Label 4-70e4-1.1-c1e2-0-12
Degree $4$
Conductor $24010000$
Sign $1$
Analytic cond. $1530.89$
Root an. cond. $6.25513$
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  + 2·3-s − 9-s − 2·11-s + 10·13-s + 10·17-s + 4·19-s − 4·23-s − 6·27-s + 2·29-s − 12·31-s − 4·33-s + 4·37-s + 20·39-s − 4·41-s − 12·43-s + 2·47-s + 20·51-s + 16·53-s + 8·57-s − 4·59-s − 16·61-s + 8·67-s − 8·69-s − 4·71-s + 16·73-s − 2·79-s − 4·81-s + ⋯
L(s)  = 1  + 1.15·3-s − 1/3·9-s − 0.603·11-s + 2.77·13-s + 2.42·17-s + 0.917·19-s − 0.834·23-s − 1.15·27-s + 0.371·29-s − 2.15·31-s − 0.696·33-s + 0.657·37-s + 3.20·39-s − 0.624·41-s − 1.82·43-s + 0.291·47-s + 2.80·51-s + 2.19·53-s + 1.05·57-s − 0.520·59-s − 2.04·61-s + 0.977·67-s − 0.963·69-s − 0.474·71-s + 1.87·73-s − 0.225·79-s − 4/9·81-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(24010000\)    =    \(2^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1530.89\)
Root analytic conductor: \(6.25513\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{4900} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 24010000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.394384459\)
\(L(\frac12)\) \(\approx\) \(5.394384459\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3$D_{4}$ \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 10 T + 49 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 - 4 T + 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 2 T + 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 12 T + 96 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 - 4 T + 76 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 + 4 T + 84 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 12 T + 90 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 16 T + 152 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 4 T + 120 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 16 T + 178 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 - 8 T + 100 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 4 T + 74 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 16 T + 202 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 2 T - 41 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 - 110 T^{2} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 6 T + 41 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
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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

−8.361251205470429203853807829585, −8.266715791654007284199784674159, −7.74753664414478560233940379182, −7.56714521937640808791754525658, −7.18736704822174484613294152477, −6.67334678594833714094697425699, −6.03747783793947854682941272301, −5.92971230632405879527100337486, −5.55664483963845982611713316925, −5.38306296317573869575170057496, −4.75545191256132167349950433369, −4.15304975336719562120775089236, −3.56787507644199132378158165433, −3.53915601125648699072611097541, −3.11357139442858685877803954465, −3.00871417529586453069695919571, −1.91982331486241435031840599837, −1.90588883351041407079731985295, −1.13248615436590951685514244052, −0.63636941224015383187025527762, 0.63636941224015383187025527762, 1.13248615436590951685514244052, 1.90588883351041407079731985295, 1.91982331486241435031840599837, 3.00871417529586453069695919571, 3.11357139442858685877803954465, 3.53915601125648699072611097541, 3.56787507644199132378158165433, 4.15304975336719562120775089236, 4.75545191256132167349950433369, 5.38306296317573869575170057496, 5.55664483963845982611713316925, 5.92971230632405879527100337486, 6.03747783793947854682941272301, 6.67334678594833714094697425699, 7.18736704822174484613294152477, 7.56714521937640808791754525658, 7.74753664414478560233940379182, 8.266715791654007284199784674159, 8.361251205470429203853807829585

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