Properties

Label 4-672e2-1.1-c1e2-0-46
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  − 3·3-s + 3·7-s + 6·9-s − 6·13-s + 19-s − 9·21-s − 25-s − 9·27-s + 7·31-s − 7·37-s + 18·39-s + 2·49-s − 3·57-s − 2·61-s + 18·63-s − 67-s − 7·73-s + 3·75-s − 7·79-s + 9·81-s − 18·91-s − 21·93-s + 12·97-s + 17·103-s − 7·109-s + 21·111-s − 36·117-s + ⋯
L(s)  = 1  − 1.73·3-s + 1.13·7-s + 2·9-s − 1.66·13-s + 0.229·19-s − 1.96·21-s − 1/5·25-s − 1.73·27-s + 1.25·31-s − 1.15·37-s + 2.88·39-s + 2/7·49-s − 0.397·57-s − 0.256·61-s + 2.26·63-s − 0.122·67-s − 0.819·73-s + 0.346·75-s − 0.787·79-s + 81-s − 1.88·91-s − 2.17·93-s + 1.21·97-s + 1.67·103-s − 0.670·109-s + 1.99·111-s − 3.32·117-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$C_2$ \( 1 + p T + p T^{2} \)
7$C_2$ \( 1 - 3 T + p T^{2} \)
good5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2^2$ \( 1 + 17 T^{2} + p^{2} T^{4} \)
13$C_2$$\times$$C_2$ \( ( 1 + T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
17$C_2^2$ \( 1 + 5 T^{2} + p^{2} T^{4} \)
19$C_2$$\times$$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
23$C_2^2$ \( 1 - 19 T^{2} + p^{2} T^{4} \)
29$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
31$C_2$$\times$$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + p T^{2} ) \)
37$C_2$$\times$$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
41$C_2^2$ \( 1 + 26 T^{2} + p^{2} T^{4} \)
43$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
47$C_2^2$ \( 1 + 65 T^{2} + p^{2} T^{4} \)
53$C_2^2$ \( 1 - 7 T^{2} + p^{2} T^{4} \)
59$C_2^2$ \( 1 + 77 T^{2} + p^{2} T^{4} \)
61$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
67$C_2$$\times$$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
71$C_2^2$ \( 1 + 90 T^{2} + p^{2} T^{4} \)
73$C_2$$\times$$C_2$ \( ( 1 + 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
79$C_2$$\times$$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
83$C_2^2$ \( 1 + 50 T^{2} + p^{2} T^{4} \)
89$C_2^2$ \( 1 + 73 T^{2} + p^{2} T^{4} \)
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

−8.277874377956488697303012697109, −7.71410731117866147437499698865, −7.35450248143790297800467637114, −6.99607528598158030816254869546, −6.43932913120193562062280154793, −5.97548267298282443053485818395, −5.38863418128718444736954436328, −5.09638095647658878650608673722, −4.58526120228103052961464181371, −4.41585586689347084490462347489, −3.48408447439410775348355100635, −2.59838524630422362465087088833, −1.87278980973687933067061252334, −1.10502185812500183932077865075, 0, 1.10502185812500183932077865075, 1.87278980973687933067061252334, 2.59838524630422362465087088833, 3.48408447439410775348355100635, 4.41585586689347084490462347489, 4.58526120228103052961464181371, 5.09638095647658878650608673722, 5.38863418128718444736954436328, 5.97548267298282443053485818395, 6.43932913120193562062280154793, 6.99607528598158030816254869546, 7.35450248143790297800467637114, 7.71410731117866147437499698865, 8.277874377956488697303012697109

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