Properties

Label 4-840e2-1.1-c1e2-0-38
Degree $4$
Conductor $705600$
Sign $-1$
Analytic cond. $44.9896$
Root an. cond. $2.58987$
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  − 2·2-s + 3-s + 2·4-s − 2·5-s − 2·6-s − 2·9-s + 4·10-s + 2·12-s − 2·15-s − 4·16-s + 4·18-s − 4·20-s − 25-s − 5·27-s + 4·30-s − 4·31-s + 8·32-s − 4·36-s + 4·45-s − 4·48-s − 49-s + 2·50-s + 12·53-s + 10·54-s − 4·60-s + 8·62-s − 8·64-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s − 0.894·5-s − 0.816·6-s − 2/3·9-s + 1.26·10-s + 0.577·12-s − 0.516·15-s − 16-s + 0.942·18-s − 0.894·20-s − 1/5·25-s − 0.962·27-s + 0.730·30-s − 0.718·31-s + 1.41·32-s − 2/3·36-s + 0.596·45-s − 0.577·48-s − 1/7·49-s + 0.282·50-s + 1.64·53-s + 1.36·54-s − 0.516·60-s + 1.01·62-s − 64-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 705600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(705600\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(44.9896\)
Root analytic conductor: \(2.58987\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 705600,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + p T + p T^{2} \)
3$C_2$ \( 1 - T + p T^{2} \)
5$C_2$ \( 1 + 2 T + p T^{2} \)
7$C_2$ \( 1 + T^{2} \)
good11$C_2^2$ \( 1 - 13 T^{2} + p^{2} T^{4} \)
13$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
17$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
29$C_2^2$ \( 1 - 33 T^{2} + p^{2} T^{4} \)
31$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
59$C_2^2$ \( 1 - 18 T^{2} + p^{2} T^{4} \)
61$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
67$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
83$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 + p T^{2} )^{2} \)
97$C_2^2$ \( 1 - 145 T^{2} + p^{2} T^{4} \)
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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

−8.087603252825030825454636123237, −7.85090293890936197901505599254, −7.37169265619738966914624232776, −7.12687213416235257491801163420, −6.46018095183604781320211722199, −5.97605287256255819865787795231, −5.36990522153821285718601593292, −4.81896397167360608395712600546, −4.17544475261394474161423450299, −3.67951146709553747460997942187, −3.19773560747257360737020214706, −2.37900035147794354357639857446, −1.96935921425456951578292518269, −0.917061660470196146908124227359, 0, 0.917061660470196146908124227359, 1.96935921425456951578292518269, 2.37900035147794354357639857446, 3.19773560747257360737020214706, 3.67951146709553747460997942187, 4.17544475261394474161423450299, 4.81896397167360608395712600546, 5.36990522153821285718601593292, 5.97605287256255819865787795231, 6.46018095183604781320211722199, 7.12687213416235257491801163420, 7.37169265619738966914624232776, 7.85090293890936197901505599254, 8.087603252825030825454636123237

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