Properties

Label 4-70e4-1.1-c1e2-0-2
Degree $4$
Conductor $24010000$
Sign $1$
Analytic cond. $1530.89$
Root an. cond. $6.25513$
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 − 6·29-s + 4·31-s + 10·41-s + 16·59-s + 14·61-s + 16·71-s − 8·79-s − 26·89-s + 12·99-s − 6·101-s − 18·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 10·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 9-s − 1.20·11-s − 1.11·29-s + 0.718·31-s + 1.56·41-s + 2.08·59-s + 1.79·61-s + 1.89·71-s − 0.900·79-s − 2.75·89-s + 1.20·99-s − 0.597·101-s − 1.72·109-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 − 0.769·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24010000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(24010000\)    =    \(2^{4} \cdot 5^{4} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(1530.89\)
Root analytic conductor: \(6.25513\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 24010000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.122010517\)
\(L(\frac12)\) \(\approx\) \(1.122010517\)
\(L(\frac{3}{2})\) 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 \( 1 \)
7 \( 1 \)
good3$C_2^2$ \( 1 + p T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 35 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 85 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 125 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 165 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
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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

−8.927606384832135505937107537569, −8.244832434877401667982860938815, −7.75819461750138587573321806343, −7.38036159906017304798779246382, −7.11633872769919272074151889212, −6.48257535512681288714544784869, −6.42463364161927056867041349096, −5.64733386520496298938240457734, −5.61406359241653358984327465588, −5.20094445439128630483935361484, −4.97783158302393580326857325548, −4.24254958085369080754719515782, −3.86902306078908157210361053440, −3.70111564415404097309701575085, −2.83743416889672640116737305572, −2.57398002177700183351243545641, −2.48718570798707084033580842034, −1.68351657563366189079604723233, −1.00397152571148135835919303822, −0.32058553311134252531728694303, 0.32058553311134252531728694303, 1.00397152571148135835919303822, 1.68351657563366189079604723233, 2.48718570798707084033580842034, 2.57398002177700183351243545641, 2.83743416889672640116737305572, 3.70111564415404097309701575085, 3.86902306078908157210361053440, 4.24254958085369080754719515782, 4.97783158302393580326857325548, 5.20094445439128630483935361484, 5.61406359241653358984327465588, 5.64733386520496298938240457734, 6.42463364161927056867041349096, 6.48257535512681288714544784869, 7.11633872769919272074151889212, 7.38036159906017304798779246382, 7.75819461750138587573321806343, 8.244832434877401667982860938815, 8.927606384832135505937107537569

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