Properties

Label 4-1280e2-1.1-c1e2-0-55
Degree $4$
Conductor $1638400$
Sign $1$
Analytic cond. $104.465$
Root an. cond. $3.19700$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 4·5-s + 6·9-s + 8·13-s + 16·15-s + 11·25-s − 4·27-s + 8·37-s + 32·39-s − 4·41-s + 12·43-s + 24·45-s + 10·49-s + 8·53-s + 32·65-s − 28·67-s − 16·71-s + 44·75-s − 32·79-s − 37·81-s + 4·83-s + 12·89-s − 20·107-s + 32·111-s + 48·117-s + 6·121-s − 16·123-s + ⋯
L(s)  = 1  + 2.30·3-s + 1.78·5-s + 2·9-s + 2.21·13-s + 4.13·15-s + 11/5·25-s − 0.769·27-s + 1.31·37-s + 5.12·39-s − 0.624·41-s + 1.82·43-s + 3.57·45-s + 10/7·49-s + 1.09·53-s + 3.96·65-s − 3.42·67-s − 1.89·71-s + 5.08·75-s − 3.60·79-s − 4.11·81-s + 0.439·83-s + 1.27·89-s − 1.93·107-s + 3.03·111-s + 4.43·117-s + 6/11·121-s − 1.44·123-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1638400\)    =    \(2^{16} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(104.465\)
Root analytic conductor: \(3.19700\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1638400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.577261508\)
\(L(\frac12)\) \(\approx\) \(8.577261508\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 4 T + p T^{2} \)
good3$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
7$C_2^2$ \( 1 - 10 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 6 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - p T^{2} )^{2} \)
19$C_2^2$ \( 1 - 22 T^{2} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - 42 T^{2} + p^{2} T^{4} \)
29$C_2^2$ \( 1 - 54 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
53$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 82 T^{2} + p^{2} T^{4} \)
79$C_2$ \( ( 1 + 16 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
97$C_2^2$ \( 1 + 62 T^{2} + p^{2} T^{4} \)
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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

−9.702448873512547699187233314861, −9.115774200879703379344269215017, −8.961545887259046685843164646129, −8.944329732012852533872773497271, −8.400367180711993083890937552543, −8.104865261834985748844516581299, −7.35893820033940340358296337314, −7.33889791434532506640505773025, −6.52157721483254482640135406635, −5.94665255851497599667597950635, −5.77039376323110168709176252833, −5.62477918740040340339592610880, −4.37629587990723114153748126098, −4.30501615326470824572477170149, −3.51051906362093501264957702027, −3.16500733785694862934791755607, −2.53907267300452578681622799370, −2.44546607486081798252542501491, −1.54513070467557192739459073335, −1.26729189908277133068136880264, 1.26729189908277133068136880264, 1.54513070467557192739459073335, 2.44546607486081798252542501491, 2.53907267300452578681622799370, 3.16500733785694862934791755607, 3.51051906362093501264957702027, 4.30501615326470824572477170149, 4.37629587990723114153748126098, 5.62477918740040340339592610880, 5.77039376323110168709176252833, 5.94665255851497599667597950635, 6.52157721483254482640135406635, 7.33889791434532506640505773025, 7.35893820033940340358296337314, 8.104865261834985748844516581299, 8.400367180711993083890937552543, 8.944329732012852533872773497271, 8.961545887259046685843164646129, 9.115774200879703379344269215017, 9.702448873512547699187233314861

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