Properties

Label 4-7200e2-1.1-c1e2-0-27
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  + 4·13-s + 4·17-s − 12·29-s + 20·37-s − 4·41-s − 6·49-s + 12·53-s − 4·61-s + 12·73-s − 20·89-s − 4·97-s + 4·101-s − 36·109-s + 4·113-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 14·169-s + 173-s + 179-s + ⋯
L(s)  = 1  + 1.10·13-s + 0.970·17-s − 2.22·29-s + 3.28·37-s − 0.624·41-s − 6/7·49-s + 1.64·53-s − 0.512·61-s + 1.40·73-s − 2.11·89-s − 0.406·97-s + 0.398·101-s − 3.44·109-s + 0.376·113-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 − 1.07·169-s + 0.0760·173-s + 0.0747·179-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

\(L(1)\) \(\approx\) \(3.067066103\)
\(L(\frac12)\) \(\approx\) \(3.067066103\)
\(L(\frac{3}{2})\) 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$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2^2$ \( 1 + 38 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
37$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
47$C_2^2$ \( 1 + 86 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \)
71$C_2^2$ \( 1 + 110 T^{2} + p^{2} T^{4} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
79$C_2^2$ \( 1 + 30 T^{2} + p^{2} T^{4} \)
83$C_2^2$ \( 1 + 158 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−7.963679046920402532264407763238, −7.888741344818032251001908181943, −7.29032545485371000584885007411, −7.28184996764748464837294750971, −6.51712836887501865120675784078, −6.46493350809301987933909227990, −5.91460594466987383152551432696, −5.67951365770390818258446695246, −5.37717874735943276420743381115, −5.04445240460816739377206551611, −4.26654561517935832143389712215, −4.24128894933764796336293531791, −3.74614386642154963353639196215, −3.46676984338734785644367617634, −2.80438642072844496316530101248, −2.69063590046686880840196279489, −1.91656597738313881477065527347, −1.56877042670706713648182220266, −1.01382778911586294831395181025, −0.48183596480629773473459067339, 0.48183596480629773473459067339, 1.01382778911586294831395181025, 1.56877042670706713648182220266, 1.91656597738313881477065527347, 2.69063590046686880840196279489, 2.80438642072844496316530101248, 3.46676984338734785644367617634, 3.74614386642154963353639196215, 4.24128894933764796336293531791, 4.26654561517935832143389712215, 5.04445240460816739377206551611, 5.37717874735943276420743381115, 5.67951365770390818258446695246, 5.91460594466987383152551432696, 6.46493350809301987933909227990, 6.51712836887501865120675784078, 7.28184996764748464837294750971, 7.29032545485371000584885007411, 7.888741344818032251001908181943, 7.963679046920402532264407763238

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