Properties

Label 4-2880e2-1.1-c1e2-0-5
Degree $4$
Conductor $8294400$
Sign $1$
Analytic cond. $528.858$
Root an. cond. $4.79550$
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  + 4·5-s − 8·11-s − 8·19-s + 11·25-s + 8·29-s − 16·41-s − 2·49-s − 32·55-s − 24·59-s − 4·61-s − 16·71-s + 16·79-s − 32·95-s − 24·101-s + 36·109-s + 26·121-s + 24·125-s + 127-s + 131-s + 137-s + 139-s + 32·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
L(s)  = 1  + 1.78·5-s − 2.41·11-s − 1.83·19-s + 11/5·25-s + 1.48·29-s − 2.49·41-s − 2/7·49-s − 4.31·55-s − 3.12·59-s − 0.512·61-s − 1.89·71-s + 1.80·79-s − 3.28·95-s − 2.38·101-s + 3.44·109-s + 2.36·121-s + 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.65·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8294400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(8294400\)    =    \(2^{12} \cdot 3^{4} \cdot 5^{2}\)
Sign: $1$
Analytic conductor: \(528.858\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 8294400,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.284387206\)
\(L(\frac12)\) \(\approx\) \(1.284387206\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 - 4 T + p T^{2} \)
good7$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 2 T^{2} + p^{2} T^{4} \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2^2$ \( 1 - 30 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2^2$ \( 1 - 58 T^{2} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
59$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
67$C_2^2$ \( 1 - 70 T^{2} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2^2$ \( 1 - 102 T^{2} + p^{2} T^{4} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
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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

−8.976881132906237701075063963154, −8.517356484430811404139132050443, −8.361911238898810357832996204318, −7.82870986421025955430773327107, −7.55864152310844591720518129492, −6.99924103207240606696081648212, −6.48553022959796100320424243072, −6.26292399535555794204698393116, −6.05066934532300952713447646295, −5.41615957774651638031806572974, −5.13166966142863002156135050653, −4.71348060786869859378135360819, −4.62834160423863715804485224963, −3.72714178620378315688573081092, −3.08638206301047369524495091571, −2.61610636138377341604922050025, −2.56449096915662537423833706203, −1.70603665243573889886471381330, −1.58749129460359509655400923930, −0.32877874993235212148822958677, 0.32877874993235212148822958677, 1.58749129460359509655400923930, 1.70603665243573889886471381330, 2.56449096915662537423833706203, 2.61610636138377341604922050025, 3.08638206301047369524495091571, 3.72714178620378315688573081092, 4.62834160423863715804485224963, 4.71348060786869859378135360819, 5.13166966142863002156135050653, 5.41615957774651638031806572974, 6.05066934532300952713447646295, 6.26292399535555794204698393116, 6.48553022959796100320424243072, 6.99924103207240606696081648212, 7.55864152310844591720518129492, 7.82870986421025955430773327107, 8.361911238898810357832996204318, 8.517356484430811404139132050443, 8.976881132906237701075063963154

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