Properties

Label 4-586971-1.1-c1e2-0-8
Degree $4$
Conductor $586971$
Sign $-1$
Analytic cond. $37.4257$
Root an. cond. $2.47339$
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 + 2·3-s − 4-s − 4·6-s + 8·8-s + 9-s − 11-s − 2·12-s − 7·16-s − 8·17-s − 2·18-s + 2·22-s + 16·24-s − 6·25-s − 4·27-s + 12·29-s + 20·31-s − 14·32-s − 2·33-s + 16·34-s − 36-s − 12·37-s − 8·41-s + 44-s − 14·48-s + 49-s + 12·50-s + ⋯
L(s)  = 1  − 1.41·2-s + 1.15·3-s − 1/2·4-s − 1.63·6-s + 2.82·8-s + 1/3·9-s − 0.301·11-s − 0.577·12-s − 7/4·16-s − 1.94·17-s − 0.471·18-s + 0.426·22-s + 3.26·24-s − 6/5·25-s − 0.769·27-s + 2.22·29-s + 3.59·31-s − 2.47·32-s − 0.348·33-s + 2.74·34-s − 1/6·36-s − 1.97·37-s − 1.24·41-s + 0.150·44-s − 2.02·48-s + 1/7·49-s + 1.69·50-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 586971 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 586971 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(586971\)    =    \(3^{2} \cdot 7^{2} \cdot 11^{3}\)
Sign: $-1$
Analytic conductor: \(37.4257\)
Root analytic conductor: \(2.47339\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((4,\ 586971,\ (\ :1/2, 1/2),\ -1)\)

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)$
bad3$C_2$ \( 1 - 2 T + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$ \( 1 + T \)
good2$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
29$C_2$ \( ( 1 - 6 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - 10 T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 + 4 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
61$C_2$ \( ( 1 + p T^{2} )^{2} \)
67$C_2$ \( ( 1 - 8 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
83$C_2$ \( ( 1 + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
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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

−8.339912824949481407935374648788, −8.184144826454463238531498469573, −7.67378178112872223010860918103, −6.94901622138190974050877028098, −6.70583207072298392541889211635, −6.09655823991520760248083514453, −5.04842931599159887263632672015, −4.94559945005359926667122742610, −4.28286203006294711588571475853, −3.97700068246488537440823967124, −3.17103530919462504679231815837, −2.48256179865490690419904115851, −1.91939450772296709675161614006, −1.01248464275202192351966482181, 0, 1.01248464275202192351966482181, 1.91939450772296709675161614006, 2.48256179865490690419904115851, 3.17103530919462504679231815837, 3.97700068246488537440823967124, 4.28286203006294711588571475853, 4.94559945005359926667122742610, 5.04842931599159887263632672015, 6.09655823991520760248083514453, 6.70583207072298392541889211635, 6.94901622138190974050877028098, 7.67378178112872223010860918103, 8.184144826454463238531498469573, 8.339912824949481407935374648788

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