Properties

Label 4-567675-1.1-c1e2-0-0
Degree $4$
Conductor $567675$
Sign $1$
Analytic cond. $36.1954$
Root an. cond. $2.45280$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 3·4-s − 8·7-s + 9-s − 3·12-s + 12·13-s + 5·16-s + 16·19-s − 8·21-s + 25-s + 27-s + 24·28-s + 8·31-s − 3·36-s + 12·37-s + 12·39-s − 8·43-s + 5·48-s + 34·49-s − 36·52-s + 16·57-s + 12·61-s − 8·63-s − 3·64-s − 16·67-s − 12·73-s + 75-s + ⋯
L(s)  = 1  + 0.577·3-s − 3/2·4-s − 3.02·7-s + 1/3·9-s − 0.866·12-s + 3.32·13-s + 5/4·16-s + 3.67·19-s − 1.74·21-s + 1/5·25-s + 0.192·27-s + 4.53·28-s + 1.43·31-s − 1/2·36-s + 1.97·37-s + 1.92·39-s − 1.21·43-s + 0.721·48-s + 34/7·49-s − 4.99·52-s + 2.11·57-s + 1.53·61-s − 1.00·63-s − 3/8·64-s − 1.95·67-s − 1.40·73-s + 0.115·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 567675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 567675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(567675\)    =    \(3^{3} \cdot 5^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(36.1954\)
Root analytic conductor: \(2.45280\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 567675,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.555654772\)
\(L(\frac12)\) \(\approx\) \(1.555654772\)
\(L(\frac{3}{2})\) 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)$
bad3$C_1$ \( 1 - T \)
5$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
29$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 14 T + p T^{2} )^{2} \)
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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

−8.575570814640394375498951194613, −8.106022253146053953989408247460, −7.72786156373545594249219051552, −6.93833368370812223463815438834, −6.55543097381440476299846924590, −6.21947835599860291908035758339, −5.58153071011565013738124935193, −5.47662500063690060518533293616, −4.38836899822603517331769890072, −3.97599829860772318843336935157, −3.44245612858978964450827586078, −3.06884295044986510662047292336, −3.06133580739474070245483773115, −1.13897079254288435205597284658, −0.813411925921320752589004841282, 0.813411925921320752589004841282, 1.13897079254288435205597284658, 3.06133580739474070245483773115, 3.06884295044986510662047292336, 3.44245612858978964450827586078, 3.97599829860772318843336935157, 4.38836899822603517331769890072, 5.47662500063690060518533293616, 5.58153071011565013738124935193, 6.21947835599860291908035758339, 6.55543097381440476299846924590, 6.93833368370812223463815438834, 7.72786156373545594249219051552, 8.106022253146053953989408247460, 8.575570814640394375498951194613

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