Properties

Label 4-672e2-1.1-c1e2-0-39
Degree $4$
Conductor $451584$
Sign $-1$
Analytic cond. $28.7933$
Root an. cond. $2.31645$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3·7-s − 2·9-s − 6·13-s + 3·19-s + 3·21-s + 7·25-s + 5·27-s + 31-s + 5·37-s + 6·39-s − 8·43-s + 2·49-s − 3·57-s + 22·61-s + 6·63-s + 5·67-s + 73-s − 7·75-s − 17·79-s + 81-s + 18·91-s − 93-s + 12·97-s − 103-s − 11·109-s − 5·111-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.13·7-s − 2/3·9-s − 1.66·13-s + 0.688·19-s + 0.654·21-s + 7/5·25-s + 0.962·27-s + 0.179·31-s + 0.821·37-s + 0.960·39-s − 1.21·43-s + 2/7·49-s − 0.397·57-s + 2.81·61-s + 0.755·63-s + 0.610·67-s + 0.117·73-s − 0.808·75-s − 1.91·79-s + 1/9·81-s + 1.88·91-s − 0.103·93-s + 1.21·97-s − 0.0985·103-s − 1.05·109-s − 0.474·111-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: yes
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$C_2$ \( 1 + T + p T^{2} \)
7$C_2$ \( 1 + 3 T + p T^{2} \)
good5$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
11$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 - 11 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - T + p T^{2} )( 1 + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 - 2 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 10 T^{2} + p^{2} T^{4} \)
43$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
47$C_2^2$ \( 1 - 31 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 + T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 - 7 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \)
73$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 + 8 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
83$C_2^2$ \( 1 - 94 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 - 103 T^{2} + p^{2} T^{4} \)
97$C_2$$\times$$C_2$ \( ( 1 - 14 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
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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

−8.345003189241069038425144259427, −7.891485977463956497611526334652, −7.29563128178746022055830342476, −6.83128814367668579028050956782, −6.63815064709444361317617534186, −6.03943810773537879093425896765, −5.45091988678237022791731347780, −5.10958658658066901021790818529, −4.69706026158527176731437600073, −3.92723299345378771904182979240, −3.23084496472940598599232750068, −2.79464447993339900578561612890, −2.28803542447379881403124909488, −0.970113145048901789709954203376, 0, 0.970113145048901789709954203376, 2.28803542447379881403124909488, 2.79464447993339900578561612890, 3.23084496472940598599232750068, 3.92723299345378771904182979240, 4.69706026158527176731437600073, 5.10958658658066901021790818529, 5.45091988678237022791731347780, 6.03943810773537879093425896765, 6.63815064709444361317617534186, 6.83128814367668579028050956782, 7.29563128178746022055830342476, 7.891485977463956497611526334652, 8.345003189241069038425144259427

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