Properties

Label 4-728e2-1.1-c1e2-0-12
Degree $4$
Conductor $529984$
Sign $-1$
Analytic cond. $33.7922$
Root an. cond. $2.41103$
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  − 2·2-s + 2·4-s − 6·9-s − 12·11-s − 4·16-s + 8·17-s + 12·18-s + 10·19-s + 24·22-s − 25-s + 8·32-s − 16·34-s − 12·36-s − 20·38-s − 12·41-s − 2·43-s − 24·44-s + 49-s + 2·50-s + 16·59-s − 8·64-s − 12·67-s + 16·68-s − 26·73-s + 20·76-s + 27·81-s + 24·82-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 2·9-s − 3.61·11-s − 16-s + 1.94·17-s + 2.82·18-s + 2.29·19-s + 5.11·22-s − 1/5·25-s + 1.41·32-s − 2.74·34-s − 2·36-s − 3.24·38-s − 1.87·41-s − 0.304·43-s − 3.61·44-s + 1/7·49-s + 0.282·50-s + 2.08·59-s − 64-s − 1.46·67-s + 1.94·68-s − 3.04·73-s + 2.29·76-s + 3·81-s + 2.65·82-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 529984 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529984 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(529984\)    =    \(2^{6} \cdot 7^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(33.7922\)
Root analytic conductor: \(2.41103\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 529984,\ (\ :1/2, 1/2),\ -1)\)

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$C_2$ \( 1 + p T + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
13$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
good3$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
11$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
17$C_2$ \( ( 1 - 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 5 T + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
31$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
37$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
41$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
47$C_2$ \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
61$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
73$C_2$ \( ( 1 + 13 T + p T^{2} )^{2} \)
79$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
83$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 3 T + p T^{2} )^{2} \)
97$C_2$ \( ( 1 - 7 T + p T^{2} )^{2} \)
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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

−8.132853228949288843567220173884, −7.88305646625584781075065151406, −7.56547054743513962577785816857, −7.43563929744309328324545511818, −6.52485705684344802846548318760, −5.71937703930548454797341247776, −5.41741601122327886404409872235, −5.27883313770925671699270459709, −4.80985573882029724233137138017, −3.37686408847102379249227387096, −3.11744276397134147631028514628, −2.70855108917176141532124458382, −2.01494804298540122257395788992, −0.807104453335379905687125358385, 0, 0.807104453335379905687125358385, 2.01494804298540122257395788992, 2.70855108917176141532124458382, 3.11744276397134147631028514628, 3.37686408847102379249227387096, 4.80985573882029724233137138017, 5.27883313770925671699270459709, 5.41741601122327886404409872235, 5.71937703930548454797341247776, 6.52485705684344802846548318760, 7.43563929744309328324545511818, 7.56547054743513962577785816857, 7.88305646625584781075065151406, 8.132853228949288843567220173884

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