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Properties

Label 2-32340-1.1-c1-0-17
Degree $2$
Conductor $32340$
Sign $1$
Analytic cond. $258.236$
Root an. cond. $16.0697$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s + 11-s + 5·13-s − 15-s + 2·19-s − 3·23-s + 25-s + 27-s + 3·29-s + 8·31-s + 33-s + 8·37-s + 5·39-s + 3·41-s + 5·43-s − 45-s + 3·47-s − 9·53-s − 55-s + 2·57-s + 2·61-s − 5·65-s + 14·67-s − 3·69-s + 12·71-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s + 0.301·11-s + 1.38·13-s − 0.258·15-s + 0.458·19-s − 0.625·23-s + 1/5·25-s + 0.192·27-s + 0.557·29-s + 1.43·31-s + 0.174·33-s + 1.31·37-s + 0.800·39-s + 0.468·41-s + 0.762·43-s − 0.149·45-s + 0.437·47-s − 1.23·53-s − 0.134·55-s + 0.264·57-s + 0.256·61-s − 0.620·65-s + 1.71·67-s − 0.361·69-s + 1.42·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32340 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(32340\)    =    \(2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(258.236\)
Root analytic conductor: \(16.0697\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 32340,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.502392980\)
\(L(\frac12)\) \(\approx\) \(3.502392980\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 - T \)
good13 \( 1 - 5 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 - 3 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 9 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−15.05700162396073, −14.47163162518184, −13.91436246579225, −13.64328312046326, −12.98231690327253, −12.39726092859140, −11.92279750818443, −11.21974462377714, −10.93157538800762, −10.16907610826509, −9.577506611317931, −9.138538504023127, −8.384104501827344, −8.075498853221195, −7.602926571784507, −6.668801410753659, −6.362488534975674, −5.655694131550806, −4.822234413092002, −4.111154266773679, −3.769530376871230, −2.965210014633664, −2.380105400456436, −1.330027401492371, −0.7658727101620267, 0.7658727101620267, 1.330027401492371, 2.380105400456436, 2.965210014633664, 3.769530376871230, 4.111154266773679, 4.822234413092002, 5.655694131550806, 6.362488534975674, 6.668801410753659, 7.602926571784507, 8.075498853221195, 8.384104501827344, 9.138538504023127, 9.577506611317931, 10.16907610826509, 10.93157538800762, 11.21974462377714, 11.92279750818443, 12.39726092859140, 12.98231690327253, 13.64328312046326, 13.91436246579225, 14.47163162518184, 15.05700162396073

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