Properties

Label 4-1023e2-1.1-c1e2-0-0
Degree $4$
Conductor $1046529$
Sign $1$
Analytic cond. $66.7275$
Root an. cond. $2.85809$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank $0$

Origins

Origins of factors

Downloads

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 4·7-s − 2·9-s + 2·11-s − 4·16-s − 4·17-s + 4·21-s − 2·23-s − 9·25-s + 5·27-s + 7·31-s − 2·33-s + 4·48-s − 2·49-s + 4·51-s − 12·53-s + 8·63-s − 14·67-s + 2·69-s + 9·75-s − 8·77-s + 81-s − 12·83-s + 30·89-s − 7·93-s − 14·97-s − 4·99-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.51·7-s − 2/3·9-s + 0.603·11-s − 16-s − 0.970·17-s + 0.872·21-s − 0.417·23-s − 9/5·25-s + 0.962·27-s + 1.25·31-s − 0.348·33-s + 0.577·48-s − 2/7·49-s + 0.560·51-s − 1.64·53-s + 1.00·63-s − 1.71·67-s + 0.240·69-s + 1.03·75-s − 0.911·77-s + 1/9·81-s − 1.31·83-s + 3.17·89-s − 0.725·93-s − 1.42·97-s − 0.402·99-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1046529 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1046529 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(4\)
Conductor: \(1046529\)    =    \(3^{2} \cdot 11^{2} \cdot 31^{2}\)
Sign: $1$
Analytic conductor: \(66.7275\)
Root analytic conductor: \(2.85809\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((4,\ 1046529,\ (\ :1/2, 1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1670417457\)
\(L(\frac12)\) \(\approx\) \(0.1670417457\)
\(L(\frac{3}{2})\) 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$C_2$ \( 1 + T + p T^{2} \)
11$C_1$ \( ( 1 - T )^{2} \)
31$C_2$ \( 1 - 7 T + p T^{2} \)
good2$C_2$ \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \)
5$C_2$ \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \)
7$C_2$ \( ( 1 + 2 T + p T^{2} )^{2} \)
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$C_2$ \( ( 1 + p T^{2} )^{2} \)
23$C_2$ \( ( 1 + T + p T^{2} )^{2} \)
29$C_2$ \( ( 1 + p T^{2} )^{2} \)
37$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
41$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
43$C_2$ \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \)
47$C_2$ \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \)
53$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
59$C_2$ \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \)
61$C_2$ \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \)
67$C_2$ \( ( 1 + 7 T + p T^{2} )^{2} \)
71$C_2$ \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \)
73$C_2$ \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \)
79$C_2$ \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \)
83$C_2$ \( ( 1 + 6 T + p T^{2} )^{2} \)
89$C_2$ \( ( 1 - 15 T + p T^{2} )^{2} \)
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

−8.092645822972623775991227958202, −7.66063306030400773445005757831, −7.04905584286339856832475960272, −6.54752783106870958663695806946, −6.36261389471308870138602900888, −6.06486280453423135412282646885, −5.61873342620095596935827790368, −4.78406122834808387264866240936, −4.57737919460974625934417159440, −3.89505275393799069682037301141, −3.43686115300931926303337036724, −2.80997610993405345581381235480, −2.33638626306566266305246081560, −1.49172744468978291340860040305, −0.18845046518642065648728040532, 0.18845046518642065648728040532, 1.49172744468978291340860040305, 2.33638626306566266305246081560, 2.80997610993405345581381235480, 3.43686115300931926303337036724, 3.89505275393799069682037301141, 4.57737919460974625934417159440, 4.78406122834808387264866240936, 5.61873342620095596935827790368, 6.06486280453423135412282646885, 6.36261389471308870138602900888, 6.54752783106870958663695806946, 7.04905584286339856832475960272, 7.66063306030400773445005757831, 8.092645822972623775991227958202

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