Properties

Label 4-3360e2-1.1-c1e2-0-16
Degree $4$
Conductor $11289600$
Sign $1$
Analytic cond. $719.834$
Root an. cond. $5.17974$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s − 2·5-s + 2·7-s + 3·9-s − 4·11-s + 4·15-s + 4·17-s − 4·19-s − 4·21-s + 3·25-s − 4·27-s − 4·29-s + 4·31-s + 8·33-s − 4·35-s + 4·37-s + 4·41-s − 8·43-s − 6·45-s − 8·47-s + 3·49-s − 8·51-s − 8·53-s + 8·55-s + 8·57-s + 4·61-s + 6·63-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 0.755·7-s + 9-s − 1.20·11-s + 1.03·15-s + 0.970·17-s − 0.917·19-s − 0.872·21-s + 3/5·25-s − 0.769·27-s − 0.742·29-s + 0.718·31-s + 1.39·33-s − 0.676·35-s + 0.657·37-s + 0.624·41-s − 1.21·43-s − 0.894·45-s − 1.16·47-s + 3/7·49-s − 1.12·51-s − 1.09·53-s + 1.07·55-s + 1.05·57-s + 0.512·61-s + 0.755·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(11289600\)    =    \(2^{10} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(719.834\)
Root analytic conductor: \(5.17974\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3360} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 11289600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 \)
3$C_1$ \( ( 1 + T )^{2} \)
5$C_1$ \( ( 1 + T )^{2} \)
7$C_1$ \( ( 1 - T )^{2} \)
good11$D_{4}$ \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 2 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$D_{4}$ \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
41$D_{4}$ \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 110 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 74 T^{2} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
71$D_{4}$ \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 8 T + 150 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 16 T + 182 T^{2} + 16 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 32 T + 438 T^{2} - 32 p T^{3} + p^{2} T^{4} \)
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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

−8.325836126172979997024685070387, −7.918892867974253287508352781512, −7.68293300708513972525411426616, −7.52353643836255835403445424899, −6.75683819978375793390015020187, −6.74509003335604703790601718423, −5.95329657718951798907588315836, −5.93317017105576971253846273336, −5.19284306339427619044909183085, −5.12728491826308296065641623603, −4.57784708496646609264819477539, −4.40567215193202450495172573143, −3.68021303297323414908734447689, −3.51296096348529648322572089679, −2.58898585241713272153748514425, −2.48728453139799749290264788725, −1.35264141809480354806388811188, −1.31081820062107418752405489840, 0, 0, 1.31081820062107418752405489840, 1.35264141809480354806388811188, 2.48728453139799749290264788725, 2.58898585241713272153748514425, 3.51296096348529648322572089679, 3.68021303297323414908734447689, 4.40567215193202450495172573143, 4.57784708496646609264819477539, 5.12728491826308296065641623603, 5.19284306339427619044909183085, 5.93317017105576971253846273336, 5.95329657718951798907588315836, 6.74509003335604703790601718423, 6.75683819978375793390015020187, 7.52353643836255835403445424899, 7.68293300708513972525411426616, 7.918892867974253287508352781512, 8.325836126172979997024685070387

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