Properties

Label 4-2352e2-1.1-c1e2-0-37
Degree $4$
Conductor $5531904$
Sign $1$
Analytic cond. $352.718$
Root an. cond. $4.33368$
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 − 5-s + 3·9-s + 11-s + 5·13-s − 2·15-s − 8·17-s + 5·19-s − 8·23-s + 5·25-s + 4·27-s + 3·29-s + 2·31-s + 2·33-s + 3·37-s + 10·39-s + 6·41-s + 7·43-s − 3·45-s + 12·47-s − 16·51-s + 11·53-s − 55-s + 10·57-s − 5·59-s + 20·61-s − 5·65-s + ⋯
L(s)  = 1  + 1.15·3-s − 0.447·5-s + 9-s + 0.301·11-s + 1.38·13-s − 0.516·15-s − 1.94·17-s + 1.14·19-s − 1.66·23-s + 25-s + 0.769·27-s + 0.557·29-s + 0.359·31-s + 0.348·33-s + 0.493·37-s + 1.60·39-s + 0.937·41-s + 1.06·43-s − 0.447·45-s + 1.75·47-s − 2.24·51-s + 1.51·53-s − 0.134·55-s + 1.32·57-s − 0.650·59-s + 2.56·61-s − 0.620·65-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5531904 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(5531904\)    =    \(2^{8} \cdot 3^{2} \cdot 7^{4}\)
Sign: $1$
Analytic conductor: \(352.718\)
Root analytic conductor: \(4.33368\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 5531904,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.373498496\)
\(L(\frac12)\) \(\approx\) \(4.373498496\)
\(L(\frac{3}{2})\) 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$C_1$ \( ( 1 - T )^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \)
11$D_{4}$ \( 1 - T + 8 T^{2} - p T^{3} + p^{2} T^{4} \)
13$D_{4}$ \( 1 - 5 T + 18 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$D_{4}$ \( 1 - 5 T + 30 T^{2} - 5 p T^{3} + p^{2} T^{4} \)
23$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
29$D_{4}$ \( 1 - 3 T + 46 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
37$D_{4}$ \( 1 - 3 T + 62 T^{2} - 3 p T^{3} + p^{2} T^{4} \)
41$D_{4}$ \( 1 - 6 T + 34 T^{2} - 6 p T^{3} + p^{2} T^{4} \)
43$D_{4}$ \( 1 - 7 T + 84 T^{2} - 7 p T^{3} + p^{2} T^{4} \)
47$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
53$D_{4}$ \( 1 - 11 T + 122 T^{2} - 11 p T^{3} + p^{2} T^{4} \)
59$D_{4}$ \( 1 + 5 T + 110 T^{2} + 5 p T^{3} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
67$D_{4}$ \( 1 + 7 T + 132 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
73$D_{4}$ \( 1 + T + 132 T^{2} + p T^{3} + p^{2} T^{4} \)
79$D_{4}$ \( 1 + 8 T + 117 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$D_{4}$ \( 1 + 7 T + 164 T^{2} + 7 p T^{3} + p^{2} T^{4} \)
89$D_{4}$ \( 1 + 6 T + 130 T^{2} + 6 p T^{3} + p^{2} T^{4} \)
97$D_{4}$ \( 1 - 25 T + 336 T^{2} - 25 p T^{3} + p^{2} T^{4} \)
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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

−8.925345769560442873811733687169, −8.756221422934994473736749042960, −8.569171209870324155581028106034, −8.166180394087960036210291327892, −7.52871139865973799668617648835, −7.46139915203099466396060936431, −6.93975492941113296083051851753, −6.57354865071375793469797820170, −6.02166765706098272470177766319, −5.83354905636171862578522735052, −5.21516960061938349518808959414, −4.42208277536691642496296553561, −4.24119908674938175549348595859, −4.06050096723783892607725561974, −3.40933646247273372318857947399, −3.01280673678000768807105020125, −2.29991115020902254113480190304, −2.20143914136948729818358736769, −1.21409936242205407619998983890, −0.74145080458751363738525776509, 0.74145080458751363738525776509, 1.21409936242205407619998983890, 2.20143914136948729818358736769, 2.29991115020902254113480190304, 3.01280673678000768807105020125, 3.40933646247273372318857947399, 4.06050096723783892607725561974, 4.24119908674938175549348595859, 4.42208277536691642496296553561, 5.21516960061938349518808959414, 5.83354905636171862578522735052, 6.02166765706098272470177766319, 6.57354865071375793469797820170, 6.93975492941113296083051851753, 7.46139915203099466396060936431, 7.52871139865973799668617648835, 8.166180394087960036210291327892, 8.569171209870324155581028106034, 8.756221422934994473736749042960, 8.925345769560442873811733687169

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