Properties

Label 4-672e2-1.1-c1e2-0-80
Degree $4$
Conductor $451584$
Sign $-1$
Analytic cond. $28.7933$
Root an. cond. $2.31645$
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  + 2·3-s + 9-s + 8·11-s − 8·13-s − 16·23-s − 10·25-s − 4·27-s + 16·33-s + 20·37-s − 16·39-s − 8·47-s + 49-s − 20·59-s − 16·61-s − 32·69-s − 12·73-s − 20·75-s − 11·81-s − 4·83-s − 4·97-s + 8·99-s + 32·107-s + 20·109-s + 40·111-s − 8·117-s + 26·121-s + 127-s + ⋯
L(s)  = 1  + 1.15·3-s + 1/3·9-s + 2.41·11-s − 2.21·13-s − 3.33·23-s − 2·25-s − 0.769·27-s + 2.78·33-s + 3.28·37-s − 2.56·39-s − 1.16·47-s + 1/7·49-s − 2.60·59-s − 2.04·61-s − 3.85·69-s − 1.40·73-s − 2.30·75-s − 1.22·81-s − 0.439·83-s − 0.406·97-s + 0.804·99-s + 3.09·107-s + 1.91·109-s + 3.79·111-s − 0.739·117-s + 2.36·121-s + 0.0887·127-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 451584 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(451584\)    =    \(2^{10} \cdot 3^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(28.7933\)
Root analytic conductor: \(2.31645\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 451584,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good5$C_2$ \( ( 1 + p T^{2} )^{2} \)
11$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
13$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
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$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
53$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
59$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 + 8 T + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
71$C_2$ \( ( 1 + p T^{2} )^{2} \)
73$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 18 T + p T^{2} )( 1 + 18 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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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

−8.247827016681358338699106327510, −7.85643204119898614530616966999, −7.53632864653844024945923131798, −7.31000008746359385531868028906, −6.19321040018157050389604631782, −6.17236708003451018025878030760, −5.88296122377187582734081048905, −4.62285613804853643208241606946, −4.39654298826148920136877245563, −3.99709320456544753258230397198, −3.39651452170084757716747861999, −2.70117202411251547524653201529, −2.00516068697428410265218115685, −1.69348648530400294148534200114, 0, 1.69348648530400294148534200114, 2.00516068697428410265218115685, 2.70117202411251547524653201529, 3.39651452170084757716747861999, 3.99709320456544753258230397198, 4.39654298826148920136877245563, 4.62285613804853643208241606946, 5.88296122377187582734081048905, 6.17236708003451018025878030760, 6.19321040018157050389604631782, 7.31000008746359385531868028906, 7.53632864653844024945923131798, 7.85643204119898614530616966999, 8.247827016681358338699106327510

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