Properties

Label 4-7200e2-1.1-c1e2-0-5
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

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Normalization:  

Dirichlet series

L(s)  = 1  − 4·11-s − 8·19-s + 4·29-s + 16·31-s + 16·41-s + 10·49-s − 12·59-s + 4·61-s − 24·71-s − 16·79-s − 24·89-s − 20·101-s − 20·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 26·169-s + 173-s + 179-s + 181-s + ⋯
L(s)  = 1  − 1.20·11-s − 1.83·19-s + 0.742·29-s + 2.87·31-s + 2.49·41-s + 10/7·49-s − 1.56·59-s + 0.512·61-s − 2.84·71-s − 1.80·79-s − 2.54·89-s − 1.99·101-s − 1.91·109-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 + 2·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-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

\(L(1)\) \(\approx\) \(0.5974690829\)
\(L(\frac12)\) \(\approx\) \(0.5974690829\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 142 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 150 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
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

−8.263482519997862581872838225219, −7.72774562402193866270981461651, −7.60156520541526534678197450761, −6.86359202801641736529325105487, −6.80210727347003648801095748543, −6.38024546511679715111319236614, −5.95702638777339623935685508467, −5.62423476445626674947342543085, −5.48792836538713782322122496049, −4.64753055910622733873644010225, −4.57188879825516906831230876108, −4.17684028632770771531159683527, −4.05875094117863229498805118068, −3.12777500175668417183418102488, −2.81002305177037998925185578590, −2.43824364946363017935671819789, −2.41817320234858887102898167000, −1.27450346935808381323749123661, −1.22641648879376721981641133002, −0.19504476731858127944305836350, 0.19504476731858127944305836350, 1.22641648879376721981641133002, 1.27450346935808381323749123661, 2.41817320234858887102898167000, 2.43824364946363017935671819789, 2.81002305177037998925185578590, 3.12777500175668417183418102488, 4.05875094117863229498805118068, 4.17684028632770771531159683527, 4.57188879825516906831230876108, 4.64753055910622733873644010225, 5.48792836538713782322122496049, 5.62423476445626674947342543085, 5.95702638777339623935685508467, 6.38024546511679715111319236614, 6.80210727347003648801095748543, 6.86359202801641736529325105487, 7.60156520541526534678197450761, 7.72774562402193866270981461651, 8.263482519997862581872838225219

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