Properties

Label 4-4560e2-1.1-c1e2-0-11
Degree $4$
Conductor $20793600$
Sign $1$
Analytic cond. $1325.81$
Root an. cond. $6.03421$
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 − 6·11-s − 2·13-s − 4·15-s − 2·19-s − 4·21-s + 3·25-s − 4·27-s + 6·29-s − 4·31-s + 12·33-s + 4·35-s − 2·37-s + 4·39-s + 6·41-s + 2·43-s + 6·45-s − 8·49-s − 12·53-s − 12·55-s + 4·57-s − 12·59-s − 20·61-s + 6·63-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.894·5-s + 0.755·7-s + 9-s − 1.80·11-s − 0.554·13-s − 1.03·15-s − 0.458·19-s − 0.872·21-s + 3/5·25-s − 0.769·27-s + 1.11·29-s − 0.718·31-s + 2.08·33-s + 0.676·35-s − 0.328·37-s + 0.640·39-s + 0.937·41-s + 0.304·43-s + 0.894·45-s − 8/7·49-s − 1.64·53-s − 1.61·55-s + 0.529·57-s − 1.56·59-s − 2.56·61-s + 0.755·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20793600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(20793600\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 19^{2}\)
Sign: $1$
Analytic conductor: \(1325.81\)
Root analytic conductor: \(6.03421\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 20793600,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 - T )^{2} \)
19$C_1$ \( ( 1 + T )^{2} \)
good7$D_{4}$ \( 1 - 2 T + 12 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 + 6 T + 28 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 + 2 T + 24 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 34 T^{2} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 40 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T + 48 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 88 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 2 T + 60 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 82 T^{2} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 12 T + 130 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 12 T + 142 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 + 20 T + 210 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 + 20 T + 198 T^{2} + 20 p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 - 8 T + 126 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 - 12 T + 154 T^{2} - 12 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 - 18 T + 256 T^{2} - 18 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 + 2 T + 168 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
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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

−7.950953049318770412735907912873, −7.71100449868886700597569809564, −7.33881018523260123913871389899, −7.32334924772435302710126925975, −6.39156163684891117821971290387, −6.24908694924913266206824850002, −5.98373620210564573161353804353, −5.67209228302257344780254716414, −5.03278579845189155581088427552, −4.89209556028868913419168897654, −4.51342957722350334999320583339, −4.50892939430143810019069048860, −3.31228133638644354575054727177, −3.22188625947209696764156220730, −2.45703976040872182094071098321, −2.24872077202480332877347288775, −1.45679254633042410870096967824, −1.31177973225248355716976267477, 0, 0, 1.31177973225248355716976267477, 1.45679254633042410870096967824, 2.24872077202480332877347288775, 2.45703976040872182094071098321, 3.22188625947209696764156220730, 3.31228133638644354575054727177, 4.50892939430143810019069048860, 4.51342957722350334999320583339, 4.89209556028868913419168897654, 5.03278579845189155581088427552, 5.67209228302257344780254716414, 5.98373620210564573161353804353, 6.24908694924913266206824850002, 6.39156163684891117821971290387, 7.32334924772435302710126925975, 7.33881018523260123913871389899, 7.71100449868886700597569809564, 7.950953049318770412735907912873

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