Properties

Label 4-159e2-1.1-c1e2-0-0
Degree $4$
Conductor $25281$
Sign $1$
Analytic cond. $1.61193$
Root an. cond. $1.12677$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 3·4-s − 8·7-s + 6·9-s + 9·12-s − 6·13-s + 5·16-s − 10·19-s + 24·21-s − 10·25-s − 9·27-s + 24·28-s + 8·31-s − 18·36-s + 10·37-s + 18·39-s − 4·43-s − 15·48-s + 34·49-s + 18·52-s + 30·57-s − 16·61-s − 48·63-s − 3·64-s − 24·67-s − 8·73-s + 30·75-s + ⋯
L(s)  = 1  − 1.73·3-s − 3/2·4-s − 3.02·7-s + 2·9-s + 2.59·12-s − 1.66·13-s + 5/4·16-s − 2.29·19-s + 5.23·21-s − 2·25-s − 1.73·27-s + 4.53·28-s + 1.43·31-s − 3·36-s + 1.64·37-s + 2.88·39-s − 0.609·43-s − 2.16·48-s + 34/7·49-s + 2.49·52-s + 3.97·57-s − 2.04·61-s − 6.04·63-s − 3/8·64-s − 2.93·67-s − 0.936·73-s + 3.46·75-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25281 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25281 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(25281\)    =    \(3^{2} \cdot 53^{2}\)
Sign: $1$
Analytic conductor: \(1.61193\)
Root analytic conductor: \(1.12677\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(2\)
Selberg data: \((4,\ 25281,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + p T + p T^{2} \)
53$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good2$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
5$C_2$ \( ( 1 + p T^{2} )^{2} \)
7$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
43$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
73$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
89$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
97$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
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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

−10.14893340042306283098541226584, −9.892408705237142275484595174505, −9.264238826193943461669666199356, −8.862464558256167033866226367065, −7.82523748591637884492188424336, −7.20736947999270954805514319970, −6.45717906388122506414394684799, −6.04347889941024738587947019148, −5.92862808142334588402164386388, −4.64090448275237090857077645759, −4.50862835068296220400083952922, −3.70981983719512629863100465165, −2.66207110679381732381827288941, 0, 0, 2.66207110679381732381827288941, 3.70981983719512629863100465165, 4.50862835068296220400083952922, 4.64090448275237090857077645759, 5.92862808142334588402164386388, 6.04347889941024738587947019148, 6.45717906388122506414394684799, 7.20736947999270954805514319970, 7.82523748591637884492188424336, 8.862464558256167033866226367065, 9.264238826193943461669666199356, 9.892408705237142275484595174505, 10.14893340042306283098541226584

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