Properties

Label 4-5520e2-1.1-c1e2-0-8
Degree $4$
Conductor $30470400$
Sign $1$
Analytic cond. $1942.81$
Root an. cond. $6.63908$
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 + 4·13-s + 4·15-s − 6·17-s − 4·19-s − 4·21-s + 2·23-s + 3·25-s − 4·27-s + 6·29-s − 10·31-s − 4·35-s − 2·37-s − 8·39-s + 6·41-s − 4·43-s − 6·45-s − 12·47-s − 11·49-s + 12·51-s + 6·53-s + 8·57-s − 6·59-s + 16·61-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s + 0.755·7-s + 9-s + 1.10·13-s + 1.03·15-s − 1.45·17-s − 0.917·19-s − 0.872·21-s + 0.417·23-s + 3/5·25-s − 0.769·27-s + 1.11·29-s − 1.79·31-s − 0.676·35-s − 0.328·37-s − 1.28·39-s + 0.937·41-s − 0.609·43-s − 0.894·45-s − 1.75·47-s − 1.57·49-s + 1.68·51-s + 0.824·53-s + 1.05·57-s − 0.781·59-s + 2.04·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 30470400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30470400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(30470400\)    =    \(2^{8} \cdot 3^{2} \cdot 5^{2} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1942.81\)
Root analytic conductor: \(6.63908\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: induced by $\chi_{5520} (1, \cdot )$
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 30470400,\ (\ :1/2, 1/2),\ 1)\)

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$C_1$ \( ( 1 + T )^{2} \)
23$C_1$ \( ( 1 - T )^{2} \)
good7$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
11$C_2^2$ \( 1 + 16 T^{2} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
17$D_{4}$ \( 1 + 6 T + 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
19$D_{4}$ \( 1 + 4 T + 36 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
29$D_{4}$ \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
31$D_{4}$ \( 1 + 10 T + 63 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 2 T + 51 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 85 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
47$D_{4}$ \( 1 + 12 T + 124 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 - 6 T + 109 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 6 T + 73 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
61$D_{4}$ \( 1 - 16 T + 132 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
67$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
71$D_{4}$ \( 1 + 6 T + 97 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
73$D_{4}$ \( 1 - 4 T + 96 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$D_{4}$ \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 24 T + 298 T^{2} + 24 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 16 T + 234 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
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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

−7.998900351165692221970364504029, −7.70186646102301411667562043034, −7.02690533959664022719426071644, −6.88614684630696585502663221729, −6.49851412208641600319588913561, −6.43319724873137140849601587825, −5.71496757545350526494324552262, −5.45613047221678302940544463000, −5.05722448009057045034167075535, −4.60763332796836225500383153260, −4.45170956927122151966525697776, −3.99294084488089109264212353206, −3.55336477302658470044256243508, −3.29316303795388708183035422721, −2.32756786166233649443248476252, −2.19968150372779865790424260110, −1.27416998067059293327661325630, −1.21209491126946288057364593563, 0, 0, 1.21209491126946288057364593563, 1.27416998067059293327661325630, 2.19968150372779865790424260110, 2.32756786166233649443248476252, 3.29316303795388708183035422721, 3.55336477302658470044256243508, 3.99294084488089109264212353206, 4.45170956927122151966525697776, 4.60763332796836225500383153260, 5.05722448009057045034167075535, 5.45613047221678302940544463000, 5.71496757545350526494324552262, 6.43319724873137140849601587825, 6.49851412208641600319588913561, 6.88614684630696585502663221729, 7.02690533959664022719426071644, 7.70186646102301411667562043034, 7.998900351165692221970364504029

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