Properties

Label 4-2352e2-1.1-c1e2-0-25
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  − 3-s − 4·5-s − 4·11-s + 8·13-s + 4·15-s + 4·19-s + 5·25-s + 27-s + 4·29-s + 8·31-s + 4·33-s + 6·37-s − 8·39-s − 8·43-s − 8·47-s + 10·53-s + 16·55-s − 4·57-s + 4·59-s + 4·61-s − 32·65-s + 4·67-s − 16·71-s + 16·73-s − 5·75-s − 8·79-s − 81-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.78·5-s − 1.20·11-s + 2.21·13-s + 1.03·15-s + 0.917·19-s + 25-s + 0.192·27-s + 0.742·29-s + 1.43·31-s + 0.696·33-s + 0.986·37-s − 1.28·39-s − 1.21·43-s − 1.16·47-s + 1.37·53-s + 2.15·55-s − 0.529·57-s + 0.520·59-s + 0.512·61-s − 3.96·65-s + 0.488·67-s − 1.89·71-s + 1.87·73-s − 0.577·75-s − 0.900·79-s − 1/9·81-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

\(L(1)\) \(\approx\) \(1.323100193\)
\(L(\frac12)\) \(\approx\) \(1.323100193\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + T^{2} \)
7 \( 1 \)
good5$C_2^2$ \( 1 + 4 T + 11 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
11$C_2^2$ \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
17$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
19$C_2^2$ \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
23$C_2^2$ \( 1 - p T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
31$C_2^2$ \( 1 - 8 T + 33 T^{2} - 8 p T^{3} + p^{2} T^{4} \)
37$C_2^2$ \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \)
41$C_2$ \( ( 1 + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2^2$ \( 1 + 8 T + 17 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \)
59$C_2^2$ \( 1 - 4 T - 43 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
61$C_2^2$ \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
67$C_2^2$ \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \)
71$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
73$C_2^2$ \( 1 - 16 T + 183 T^{2} - 16 p T^{3} + p^{2} T^{4} \)
79$C_2^2$ \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
83$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
89$C_2^2$ \( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} \)
97$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
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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

−8.849302132574382314433410960842, −8.756165281871271597574942879492, −8.184494733608598540122512005964, −8.162454909400784836679764248996, −7.61646911045773886138778279270, −7.52295528098007240532414033495, −6.73633751379849208045691791799, −6.43996978211517577958980627708, −6.23992432093177523133383653965, −5.47846004602496441929605149751, −5.31909959994390893683180266514, −4.81348148381500157600164431427, −4.18944362206715009418096605272, −4.06688952568417034924405830656, −3.39194595247321576392358833540, −3.17957780826827283594965902667, −2.62512407886809185911565219718, −1.75270625952769023313930536336, −0.906666521490556067540907519756, −0.55750583794809480755232200728, 0.55750583794809480755232200728, 0.906666521490556067540907519756, 1.75270625952769023313930536336, 2.62512407886809185911565219718, 3.17957780826827283594965902667, 3.39194595247321576392358833540, 4.06688952568417034924405830656, 4.18944362206715009418096605272, 4.81348148381500157600164431427, 5.31909959994390893683180266514, 5.47846004602496441929605149751, 6.23992432093177523133383653965, 6.43996978211517577958980627708, 6.73633751379849208045691791799, 7.52295528098007240532414033495, 7.61646911045773886138778279270, 8.162454909400784836679764248996, 8.184494733608598540122512005964, 8.756165281871271597574942879492, 8.849302132574382314433410960842

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