Properties

Label 4-84672-1.1-c1e2-0-38
Degree $4$
Conductor $84672$
Sign $-1$
Analytic cond. $5.39876$
Root an. cond. $1.52431$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $1$

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 2·7-s + 9-s − 4·13-s − 8·19-s − 2·21-s − 6·25-s + 27-s − 16·31-s − 20·37-s − 4·39-s + 24·43-s + 3·49-s − 8·57-s − 20·61-s − 2·63-s + 24·67-s + 4·73-s − 6·75-s + 16·79-s + 81-s + 8·91-s − 16·93-s + 20·97-s + 28·109-s − 20·111-s − 4·117-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.755·7-s + 1/3·9-s − 1.10·13-s − 1.83·19-s − 0.436·21-s − 6/5·25-s + 0.192·27-s − 2.87·31-s − 3.28·37-s − 0.640·39-s + 3.65·43-s + 3/7·49-s − 1.05·57-s − 2.56·61-s − 0.251·63-s + 2.93·67-s + 0.468·73-s − 0.692·75-s + 1.80·79-s + 1/9·81-s + 0.838·91-s − 1.65·93-s + 2.03·97-s + 2.68·109-s − 1.89·111-s − 0.369·117-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 84672 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84672 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(84672\)    =    \(2^{6} \cdot 3^{3} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(5.39876\)
Root analytic conductor: \(1.52431\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 84672,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad2 \( 1 \)
3$C_1$ \( 1 - T \)
7$C_1$ \( ( 1 + T )^{2} \)
good5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
11$C_2$ \( ( 1 + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
19$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
31$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
61$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 12 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
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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

−9.376196260318872658947991042154, −8.825162906132063336486925508928, −8.820314983809012661425051705328, −7.65994621019403063529610936786, −7.61375311560988625300876494705, −6.99493796978412617912527979495, −6.42136658293515809263418102895, −5.82547356011864879754705597363, −5.25019213638990204722565722077, −4.54479169229856620879036583537, −3.69547450872939490185536964424, −3.56921408081238054859871707771, −2.27244908620345358837812210269, −2.06515692945354055956624655608, 0, 2.06515692945354055956624655608, 2.27244908620345358837812210269, 3.56921408081238054859871707771, 3.69547450872939490185536964424, 4.54479169229856620879036583537, 5.25019213638990204722565722077, 5.82547356011864879754705597363, 6.42136658293515809263418102895, 6.99493796978412617912527979495, 7.61375311560988625300876494705, 7.65994621019403063529610936786, 8.820314983809012661425051705328, 8.825162906132063336486925508928, 9.376196260318872658947991042154

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