Properties

Label 4-7200e2-1.1-c1e2-0-44
Degree $4$
Conductor $51840000$
Sign $1$
Analytic cond. $3305.36$
Root an. cond. $7.58236$
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  + 8·19-s − 4·29-s + 8·31-s − 4·41-s + 14·49-s + 16·59-s − 4·61-s − 16·71-s + 24·79-s − 28·89-s + 20·101-s + 20·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + 179-s + 181-s + 191-s + ⋯
L(s)  = 1  + 1.83·19-s − 0.742·29-s + 1.43·31-s − 0.624·41-s + 2·49-s + 2.08·59-s − 0.512·61-s − 1.89·71-s + 2.70·79-s − 2.96·89-s + 1.99·101-s + 1.91·109-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51840000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(51840000\)    =    \(2^{10} \cdot 3^{4} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(3305.36\)
Root analytic conductor: \(7.58236\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 51840000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.840137684\)
\(L(\frac12)\) \(\approx\) \(3.840137684\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 \)
good7$C_2$ \( ( 1 - p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 190 T^{2} + p^{2} T^{4} \)
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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

−7.923932089792062189652073376034, −7.80167951805214487445448134445, −7.33225346048779419575682220050, −7.22148408476620081880608888230, −6.60218657023607180623347603209, −6.50046226698284977619425210193, −5.97733570521886341192426614439, −5.51450596492615430898516928190, −5.25613984364538224159506795073, −5.17147657418041847102297016203, −4.32759556527766450302584513995, −4.27845317651902692622930911042, −3.77215898518822284684546202926, −3.30910408569666628188173587970, −2.85445161857989997352965804364, −2.70112079267375059466299038404, −1.91414046544617496136066687496, −1.64767714289116075478053708119, −0.808547064373555332679309238201, −0.64565281169397465826476935697, 0.64565281169397465826476935697, 0.808547064373555332679309238201, 1.64767714289116075478053708119, 1.91414046544617496136066687496, 2.70112079267375059466299038404, 2.85445161857989997352965804364, 3.30910408569666628188173587970, 3.77215898518822284684546202926, 4.27845317651902692622930911042, 4.32759556527766450302584513995, 5.17147657418041847102297016203, 5.25613984364538224159506795073, 5.51450596492615430898516928190, 5.97733570521886341192426614439, 6.50046226698284977619425210193, 6.60218657023607180623347603209, 7.22148408476620081880608888230, 7.33225346048779419575682220050, 7.80167951805214487445448134445, 7.923932089792062189652073376034

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