Properties

Label 4-2300e2-1.1-c1e2-0-3
Degree $4$
Conductor $5290000$
Sign $1$
Analytic cond. $337.294$
Root an. cond. $4.28550$
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  − 3·9-s + 8·19-s − 2·29-s + 2·31-s + 22·41-s + 10·49-s + 16·59-s − 16·61-s + 26·71-s + 24·79-s + 12·89-s − 28·101-s − 4·109-s − 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 17·169-s − 24·171-s + 173-s + 179-s + ⋯
L(s)  = 1  − 9-s + 1.83·19-s − 0.371·29-s + 0.359·31-s + 3.43·41-s + 10/7·49-s + 2.08·59-s − 2.04·61-s + 3.08·71-s + 2.70·79-s + 1.27·89-s − 2.78·101-s − 0.383·109-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 + 1.30·169-s − 1.83·171-s + 0.0760·173-s + 0.0747·179-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5290000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5290000\)    =    \(2^{4} \cdot 5^{4} \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(337.294\)
Root analytic conductor: \(4.28550\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5290000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.576642971\)
\(L(\frac12)\) \(\approx\) \(2.576642971\)
\(L(\frac{3}{2})\) 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 \( 1 \)
23$C_2$ \( 1 + T^{2} \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 17 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 11 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 93 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 13 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 97 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
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

−9.421386391757468799535425724509, −8.925584223107731586562249457510, −8.415521752186210942554337934259, −7.88461422588349372054447237224, −7.67054414773202156090740228011, −7.55762096548734733721171084235, −6.71998876418262192521442454433, −6.57368638330987671878358124317, −6.04917306573292468589968168399, −5.52765018458840601001632351549, −5.32196017541872332548137792984, −5.06767182715980475100531215196, −4.18934621861816098594634005406, −3.98708815456757860392562969740, −3.48079896726348622349116182113, −2.74217315865621733906150214856, −2.67736924139985412003182190816, −2.00504511860038392074354318627, −1.04302265877980714820041650158, −0.66666794379353822211189918673, 0.66666794379353822211189918673, 1.04302265877980714820041650158, 2.00504511860038392074354318627, 2.67736924139985412003182190816, 2.74217315865621733906150214856, 3.48079896726348622349116182113, 3.98708815456757860392562969740, 4.18934621861816098594634005406, 5.06767182715980475100531215196, 5.32196017541872332548137792984, 5.52765018458840601001632351549, 6.04917306573292468589968168399, 6.57368638330987671878358124317, 6.71998876418262192521442454433, 7.55762096548734733721171084235, 7.67054414773202156090740228011, 7.88461422588349372054447237224, 8.415521752186210942554337934259, 8.925584223107731586562249457510, 9.421386391757468799535425724509

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