Properties

Label 4-586971-1.1-c1e2-0-9
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 $2$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 4·4-s + 6·9-s + 11-s + 12·12-s + 12·16-s − 4·17-s − 9·25-s − 9·27-s − 20·29-s + 2·31-s − 3·33-s − 24·36-s − 10·37-s + 4·41-s − 4·44-s − 36·48-s + 49-s + 12·51-s − 32·64-s − 6·67-s + 16·68-s + 27·75-s + 9·81-s − 24·83-s + 60·87-s − 6·93-s + ⋯
L(s)  = 1  − 1.73·3-s − 2·4-s + 2·9-s + 0.301·11-s + 3.46·12-s + 3·16-s − 0.970·17-s − 9/5·25-s − 1.73·27-s − 3.71·29-s + 0.359·31-s − 0.522·33-s − 4·36-s − 1.64·37-s + 0.624·41-s − 0.603·44-s − 5.19·48-s + 1/7·49-s + 1.68·51-s − 4·64-s − 0.733·67-s + 1.94·68-s + 3.11·75-s + 81-s − 2.63·83-s + 6.43·87-s − 0.622·93-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: \(2\)
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$C_2$ \( 1 + p T + p T^{2} \)
7$C_1$$\times$$C_1$ \( ( 1 - T )( 1 + T ) \)
11$C_1$ \( 1 - T \)
good2$C_2$ \( ( 1 + p T^{2} )^{2} \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
13$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
17$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
19$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
23$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
29$C_2$ \( ( 1 + 10 T + p T^{2} )^{2} \)
31$C_2$ \( ( 1 - T + p T^{2} )^{2} \)
37$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
41$C_2$ \( ( 1 - 2 T + p T^{2} )^{2} \)
43$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
59$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 3 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
73$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 12 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \)
97$C_2$ \( ( 1 + 5 T + p T^{2} )^{2} \)
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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

−7.956327480837723148155835866573, −7.41775134048211629573821110448, −7.17057178520211720164419084101, −6.42370236549041572726553632076, −5.80118601537640384820432803021, −5.58596676881303008604275522263, −5.38913047752314397636287568574, −4.53418395180193611114078817717, −4.39514744576497898862871693801, −3.79024025385094143595808125256, −3.48522622489004363314719629746, −2.00735365908684561839702035924, −1.32889476078570135837429482480, 0, 0, 1.32889476078570135837429482480, 2.00735365908684561839702035924, 3.48522622489004363314719629746, 3.79024025385094143595808125256, 4.39514744576497898862871693801, 4.53418395180193611114078817717, 5.38913047752314397636287568574, 5.58596676881303008604275522263, 5.80118601537640384820432803021, 6.42370236549041572726553632076, 7.17057178520211720164419084101, 7.41775134048211629573821110448, 7.956327480837723148155835866573

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