Properties

Label 4-3360e2-1.1-c1e2-0-5
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 $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4·5-s + 3·9-s + 4·13-s − 8·15-s + 11·25-s + 4·27-s + 8·31-s − 12·37-s + 8·39-s + 20·43-s − 12·45-s − 49-s − 28·53-s − 16·65-s − 20·67-s + 24·71-s + 22·75-s + 28·79-s + 5·81-s + 24·83-s + 16·89-s + 16·93-s + 8·107-s − 24·111-s + 12·117-s + 6·121-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s + 9-s + 1.10·13-s − 2.06·15-s + 11/5·25-s + 0.769·27-s + 1.43·31-s − 1.97·37-s + 1.28·39-s + 3.04·43-s − 1.78·45-s − 1/7·49-s − 3.84·53-s − 1.98·65-s − 2.44·67-s + 2.84·71-s + 2.54·75-s + 3.15·79-s + 5/9·81-s + 2.63·83-s + 1.69·89-s + 1.65·93-s + 0.773·107-s − 2.27·111-s + 1.10·117-s + 6/11·121-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: \(0\)
Selberg data: \((4,\ 11289600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.020632629\)
\(L(\frac12)\) \(\approx\) \(3.020632629\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 4 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 118 T^{2} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 14 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 158 T^{2} + p^{2} T^{4} \)
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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

−8.567819942799249714602010885128, −8.426336635636305127068186430565, −8.098669702489921424113953329183, −7.68130236284686718342587077275, −7.55835377641132013487815061062, −7.19264089521570009848293430474, −6.48689932135419456135463669843, −6.34892621300295132023060141861, −6.05003892972168458098315915763, −5.06674516919571137340226387986, −4.81489010576000963676604277736, −4.61824952685902356533292830462, −3.78978205021579893172733102290, −3.76595964921642079534572064373, −3.41455003614794314970890911207, −2.95233662161984075967218719510, −2.41801035713232971077379923375, −1.81906811780195090324792399110, −1.10170041188640126478914951421, −0.56381971390923017164732230783, 0.56381971390923017164732230783, 1.10170041188640126478914951421, 1.81906811780195090324792399110, 2.41801035713232971077379923375, 2.95233662161984075967218719510, 3.41455003614794314970890911207, 3.76595964921642079534572064373, 3.78978205021579893172733102290, 4.61824952685902356533292830462, 4.81489010576000963676604277736, 5.06674516919571137340226387986, 6.05003892972168458098315915763, 6.34892621300295132023060141861, 6.48689932135419456135463669843, 7.19264089521570009848293430474, 7.55835377641132013487815061062, 7.68130236284686718342587077275, 8.098669702489921424113953329183, 8.426336635636305127068186430565, 8.567819942799249714602010885128

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