Properties

Label 4-2300e2-1.1-c1e2-0-1
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·9-s + 4·11-s + 4·19-s + 14·29-s − 6·31-s − 18·41-s − 2·49-s − 4·61-s − 6·71-s + 12·79-s − 24·89-s − 12·99-s + 4·101-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s − 12·171-s + 173-s + 179-s + ⋯
L(s)  = 1  − 9-s + 1.20·11-s + 0.917·19-s + 2.59·29-s − 1.07·31-s − 2.81·41-s − 2/7·49-s − 0.512·61-s − 0.712·71-s + 1.35·79-s − 2.54·89-s − 1.20·99-s + 0.398·101-s − 0.909·121-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 + 1/13·169-s − 0.917·171-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

\(L(1)\) \(\approx\) \(1.929990838\)
\(L(\frac12)\) \(\approx\) \(1.929990838\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 9 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 137 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - p T^{2} )^{2} \)
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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

−9.453232625719622594676587094405, −8.807934599286870039841725370961, −8.542634140209747277331368692949, −7.967223063091418866256431540680, −7.906559160547733904107573104139, −7.10443887254701301761528985561, −6.87676144920437328319180047844, −6.44401110986054027910002976974, −6.25357164178750971939517152509, −5.60371667903931449094771374673, −5.15764383244554427127128053646, −5.03855497447561532086843115276, −4.27298069604204008621161791665, −3.98828664273168491924632110052, −3.29544738885688112754211207746, −3.06351385750859568811271146788, −2.60653685873754281974076823388, −1.71020384637624063252578969762, −1.37483397753139209654325389553, −0.49407356260534879461360775407, 0.49407356260534879461360775407, 1.37483397753139209654325389553, 1.71020384637624063252578969762, 2.60653685873754281974076823388, 3.06351385750859568811271146788, 3.29544738885688112754211207746, 3.98828664273168491924632110052, 4.27298069604204008621161791665, 5.03855497447561532086843115276, 5.15764383244554427127128053646, 5.60371667903931449094771374673, 6.25357164178750971939517152509, 6.44401110986054027910002976974, 6.87676144920437328319180047844, 7.10443887254701301761528985561, 7.906559160547733904107573104139, 7.967223063091418866256431540680, 8.542634140209747277331368692949, 8.807934599286870039841725370961, 9.453232625719622594676587094405

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