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Properties

Label 2-32340-1.1-c1-0-4
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 − 4·13-s − 15-s − 2·17-s + 6·19-s − 4·23-s + 25-s + 27-s − 4·29-s − 33-s + 6·37-s − 4·39-s − 8·43-s − 45-s + 4·47-s − 2·51-s − 6·53-s + 55-s + 6·57-s − 4·59-s − 6·61-s + 4·65-s − 4·67-s − 4·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.258·15-s − 0.485·17-s + 1.37·19-s − 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s − 0.174·33-s + 0.986·37-s − 0.640·39-s − 1.21·43-s − 0.149·45-s + 0.583·47-s − 0.280·51-s − 0.824·53-s + 0.134·55-s + 0.794·57-s − 0.520·59-s − 0.768·61-s + 0.496·65-s − 0.488·67-s − 0.481·69-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

\(L(1)\) \(\approx\) \(1.661818321\)
\(L(\frac12)\) \(\approx\) \(1.661818321\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−15.03764548080697, −14.46779229101329, −14.11777234978179, −13.45579516170682, −13.00034082868861, −12.41499257825923, −11.83282740779653, −11.47090992056910, −10.76265288336820, −10.07218101608939, −9.677739900570508, −9.173445010629878, −8.495158896449551, −7.888456872691582, −7.434584586332641, −7.090399965711633, −6.171079055646045, −5.583317022027997, −4.747970670686223, −4.439743820533606, −3.500039374912174, −3.042622072526858, −2.291448663330787, −1.583973661104164, −0.4592825892552174, 0.4592825892552174, 1.583973661104164, 2.291448663330787, 3.042622072526858, 3.500039374912174, 4.439743820533606, 4.747970670686223, 5.583317022027997, 6.171079055646045, 7.090399965711633, 7.434584586332641, 7.888456872691582, 8.495158896449551, 9.173445010629878, 9.677739900570508, 10.07218101608939, 10.76265288336820, 11.47090992056910, 11.83282740779653, 12.41499257825923, 13.00034082868861, 13.45579516170682, 14.11777234978179, 14.46779229101329, 15.03764548080697

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