Properties

Label 4-3920e2-1.1-c1e2-0-10
Degree $4$
Conductor $15366400$
Sign $1$
Analytic cond. $979.774$
Root an. cond. $5.59476$
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 − 9-s − 4·11-s − 4·13-s − 4·15-s + 4·17-s + 2·23-s + 3·25-s + 6·27-s − 2·29-s + 12·31-s + 8·33-s + 8·39-s − 10·41-s − 10·43-s − 2·45-s − 4·47-s − 8·51-s − 8·53-s − 8·55-s + 8·59-s − 6·61-s − 8·65-s − 22·67-s − 4·69-s + 8·71-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s − 1/3·9-s − 1.20·11-s − 1.10·13-s − 1.03·15-s + 0.970·17-s + 0.417·23-s + 3/5·25-s + 1.15·27-s − 0.371·29-s + 2.15·31-s + 1.39·33-s + 1.28·39-s − 1.56·41-s − 1.52·43-s − 0.298·45-s − 0.583·47-s − 1.12·51-s − 1.09·53-s − 1.07·55-s + 1.04·59-s − 0.768·61-s − 0.992·65-s − 2.68·67-s − 0.481·69-s + 0.949·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(15366400\)    =    \(2^{8} \cdot 5^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(979.774\)
Root analytic conductor: \(5.59476\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{3920} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 15366400,\ (\ :1/2, 1/2),\ 1)\)

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 \( 1 \)
5$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good3$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 - 2 T + 45 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + p T^{2} )^{2} \)
41$D_{4}$ \( 1 + 10 T + 99 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 + 10 T + 109 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 - 8 T + 62 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 6 T + 59 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
67$D_{4}$ \( 1 + 22 T + 253 T^{2} + 22 p T^{3} + p^{2} T^{4} \)
71$D_{4}$ \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 4 T + 142 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 24 T + 294 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 2 T + 5 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 155 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 12 T + 198 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
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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

−8.206011324173688267594849797163, −8.011720520931870932364051601660, −7.40708693989609610217642267455, −7.23746412904578506307064904724, −6.62641534382088283647335788525, −6.42166025279623473738336051477, −5.87669923672533610265851951844, −5.81033451226306419630331086382, −5.18661549011192502982190408329, −5.02901983418390223994746671800, −4.81675106584660287909935358319, −4.33786214732182446865009859525, −3.38553297403491952468088848846, −3.19633947561842943398030785236, −2.59942774280718036414860003182, −2.43507705103961987308389020965, −1.54042571092675288000290094150, −1.17177997092388157997457356948, 0, 0, 1.17177997092388157997457356948, 1.54042571092675288000290094150, 2.43507705103961987308389020965, 2.59942774280718036414860003182, 3.19633947561842943398030785236, 3.38553297403491952468088848846, 4.33786214732182446865009859525, 4.81675106584660287909935358319, 5.02901983418390223994746671800, 5.18661549011192502982190408329, 5.81033451226306419630331086382, 5.87669923672533610265851951844, 6.42166025279623473738336051477, 6.62641534382088283647335788525, 7.23746412904578506307064904724, 7.40708693989609610217642267455, 8.011720520931870932364051601660, 8.206011324173688267594849797163

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