Properties

Label 4-3200e2-1.1-c1e2-0-4
Degree $4$
Conductor $10240000$
Sign $1$
Analytic cond. $652.911$
Root an. cond. $5.05491$
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  + 2·3-s − 2·7-s + 2·9-s − 4·11-s − 4·21-s − 14·23-s + 6·27-s + 4·29-s − 4·31-s − 8·33-s − 4·37-s + 16·41-s + 2·43-s − 10·47-s − 6·49-s − 8·53-s − 8·59-s + 12·61-s − 4·63-s − 2·67-s − 28·69-s − 4·71-s + 8·77-s − 8·79-s + 11·81-s − 6·83-s + 8·87-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.755·7-s + 2/3·9-s − 1.20·11-s − 0.872·21-s − 2.91·23-s + 1.15·27-s + 0.742·29-s − 0.718·31-s − 1.39·33-s − 0.657·37-s + 2.49·41-s + 0.304·43-s − 1.45·47-s − 6/7·49-s − 1.09·53-s − 1.04·59-s + 1.53·61-s − 0.503·63-s − 0.244·67-s − 3.37·69-s − 0.474·71-s + 0.911·77-s − 0.900·79-s + 11/9·81-s − 0.658·83-s + 0.857·87-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10240000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(10240000\)    =    \(2^{14} \cdot 5^{4}\)
Sign: $1$
Analytic conductor: \(652.911\)
Root analytic conductor: \(5.05491\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 10240000,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.247819787\)
\(L(\frac12)\) \(\approx\) \(1.247819787\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good3$C_2^2$ \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
7$D_{4}$ \( 1 + 2 T + 10 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
11$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
13$C_2^2$ \( 1 + 6 T^{2} + p^{2} T^{4} \)
17$C_2^2$ \( 1 + 14 T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 + 18 T^{2} + p^{2} T^{4} \)
23$D_{4}$ \( 1 + 14 T + 90 T^{2} + 14 p T^{3} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_4$ \( 1 + 4 T + 46 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
37$D_{4}$ \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
41$C_4$ \( 1 - 16 T + 126 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \)
47$D_{4}$ \( 1 + 10 T + 114 T^{2} + 10 p T^{3} + p^{2} T^{4} \)
53$D_{4}$ \( 1 + 8 T + 102 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
59$C_4$ \( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 2 T + 90 T^{2} + 2 p T^{3} + p^{2} T^{4} \)
71$C_4$ \( 1 + 4 T + 126 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
73$C_2^2$ \( 1 + 126 T^{2} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 94 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 6 T + 130 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
89$C_4$ \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 24 T + 318 T^{2} - 24 p T^{3} + p^{2} T^{4} \)
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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

−8.785631307212559521321069099643, −8.387920427143050887541971240537, −8.120948978448200804221875564894, −7.80645298040236734772696037493, −7.57073666810969959295563204392, −7.08474006308187817248818573949, −6.61203504211151017252583510885, −6.17619205849398881546287945266, −5.91092334352749397962602602066, −5.56638162067760993957574079007, −4.82055817428012073790651607440, −4.59640412599998918345589801117, −4.12859871426762515575576796394, −3.58455068408976506383444900884, −3.24021451352266214478012286503, −2.87515495709318607632567513210, −2.27193032079578618463059172699, −2.09240636065476640698698567071, −1.32670692859356276913868755939, −0.30619744440693170843345175037, 0.30619744440693170843345175037, 1.32670692859356276913868755939, 2.09240636065476640698698567071, 2.27193032079578618463059172699, 2.87515495709318607632567513210, 3.24021451352266214478012286503, 3.58455068408976506383444900884, 4.12859871426762515575576796394, 4.59640412599998918345589801117, 4.82055817428012073790651607440, 5.56638162067760993957574079007, 5.91092334352749397962602602066, 6.17619205849398881546287945266, 6.61203504211151017252583510885, 7.08474006308187817248818573949, 7.57073666810969959295563204392, 7.80645298040236734772696037493, 8.120948978448200804221875564894, 8.387920427143050887541971240537, 8.785631307212559521321069099643

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