Properties

Label 4-5040e2-1.1-c1e2-0-7
Degree $4$
Conductor $25401600$
Sign $1$
Analytic cond. $1619.62$
Root an. cond. $6.34386$
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  − 2·5-s − 12·11-s − 12·19-s − 25-s − 4·29-s + 20·31-s − 4·41-s − 49-s + 24·55-s + 16·59-s − 4·61-s + 20·71-s + 8·79-s + 12·89-s + 24·95-s + 12·101-s − 4·109-s + 86·121-s + 12·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s − 40·155-s + ⋯
L(s)  = 1  − 0.894·5-s − 3.61·11-s − 2.75·19-s − 1/5·25-s − 0.742·29-s + 3.59·31-s − 0.624·41-s − 1/7·49-s + 3.23·55-s + 2.08·59-s − 0.512·61-s + 2.37·71-s + 0.900·79-s + 1.27·89-s + 2.46·95-s + 1.19·101-s − 0.383·109-s + 7.81·121-s + 1.07·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s − 3.21·155-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25401600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25401600\)    =    \(2^{8} \cdot 3^{4} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1619.62\)
Root analytic conductor: \(6.34386\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 25401600,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8988936580\)
\(L(\frac12)\) \(\approx\) \(0.8988936580\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + 2 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - p T^{2} )^{2} \)
53$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 122 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
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

−8.361835118147274635800141074582, −8.019167932013457252481775158744, −7.70625668077413226833832602500, −7.69079784765016016213972563289, −6.97403708547201024351216230581, −6.63075326421363053411211455733, −6.23338218497328951643090127303, −5.99481893428776636770374903919, −5.23252694421221913589755940860, −5.21264409329943750214104591463, −4.71565485860379540356833093134, −4.46823829278461423382501114175, −3.94985970118606500879049095822, −3.57058368523390818353692304276, −2.81268314501469923363982832570, −2.75377545149421420779722922875, −2.11936001204954947106587096675, −2.05209531792832831436100079288, −0.66703873303515375556556336169, −0.38284125303428490294899336549, 0.38284125303428490294899336549, 0.66703873303515375556556336169, 2.05209531792832831436100079288, 2.11936001204954947106587096675, 2.75377545149421420779722922875, 2.81268314501469923363982832570, 3.57058368523390818353692304276, 3.94985970118606500879049095822, 4.46823829278461423382501114175, 4.71565485860379540356833093134, 5.21264409329943750214104591463, 5.23252694421221913589755940860, 5.99481893428776636770374903919, 6.23338218497328951643090127303, 6.63075326421363053411211455733, 6.97403708547201024351216230581, 7.69079784765016016213972563289, 7.70625668077413226833832602500, 8.019167932013457252481775158744, 8.361835118147274635800141074582

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