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Properties

Label 2-32340-1.1-c1-0-28
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s − 5-s + 9-s − 11-s − 3·13-s − 15-s − 2·17-s + 2·23-s + 25-s + 27-s + 6·29-s − 11·31-s − 33-s + 4·37-s − 3·39-s + 2·41-s + 11·43-s − 45-s − 6·47-s − 2·51-s − 4·53-s + 55-s + 3·59-s − 8·61-s + 3·65-s + 8·67-s + 2·69-s + ⋯
L(s)  = 1  + 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s − 0.832·13-s − 0.258·15-s − 0.485·17-s + 0.417·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s − 1.97·31-s − 0.174·33-s + 0.657·37-s − 0.480·39-s + 0.312·41-s + 1.67·43-s − 0.149·45-s − 0.875·47-s − 0.280·51-s − 0.549·53-s + 0.134·55-s + 0.390·59-s − 1.02·61-s + 0.372·65-s + 0.977·67-s + 0.240·69-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: \(1\)
Selberg data: \((2,\ 32340,\ (\ :1/2),\ -1)\)

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$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
5 \( 1 + T \)
7 \( 1 \)
11 \( 1 + T \)
good13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 11 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 3 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 - 3 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 - 3 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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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

−15.32980584100982, −14.70976281688250, −14.29736992391680, −13.91102159525055, −13.02363195883644, −12.72248577606554, −12.35549868453506, −11.46827910050938, −11.11228063046802, −10.53019581232317, −9.848531342172859, −9.372569660275561, −8.820340050477963, −8.279985157486837, −7.565855960191878, −7.331364754354535, −6.614027647830344, −5.918373985889347, −5.111526398195469, −4.627532126818653, −3.965003139428284, −3.285896529926569, −2.597120080283397, −2.042861051871305, −1.002461993011462, 0, 1.002461993011462, 2.042861051871305, 2.597120080283397, 3.285896529926569, 3.965003139428284, 4.627532126818653, 5.111526398195469, 5.918373985889347, 6.614027647830344, 7.331364754354535, 7.565855960191878, 8.279985157486837, 8.820340050477963, 9.372569660275561, 9.848531342172859, 10.53019581232317, 11.11228063046802, 11.46827910050938, 12.35549868453506, 12.72248577606554, 13.02363195883644, 13.91102159525055, 14.29736992391680, 14.70976281688250, 15.32980584100982

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