Properties

Label 2-32340-1.1-c1-0-12
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 + 7·13-s + 15-s + 2·17-s + 2·23-s + 25-s − 27-s + 2·29-s − 3·31-s + 33-s + 12·37-s − 7·39-s + 6·41-s + 43-s − 45-s − 10·47-s − 2·51-s + 55-s + 7·59-s − 7·65-s − 4·67-s − 2·69-s + 9·71-s + 9·73-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.301·11-s + 1.94·13-s + 0.258·15-s + 0.485·17-s + 0.417·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.538·31-s + 0.174·33-s + 1.97·37-s − 1.12·39-s + 0.937·41-s + 0.152·43-s − 0.149·45-s − 1.45·47-s − 0.280·51-s + 0.134·55-s + 0.911·59-s − 0.868·65-s − 0.488·67-s − 0.240·69-s + 1.06·71-s + 1.05·73-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: $\chi_{32340} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 32340,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.127938051\)
\(L(\frac12)\) \(\approx\) \(2.127938051\)
\(L(\frac{3}{2})\) 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 \( 1 - 7 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 3 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 7 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 9 T + p T^{2} \)
73 \( 1 - 9 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 11 T + p T^{2} \)
89 \( 1 - 7 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−15.09643309124633, −14.57552609496652, −13.98565083309624, −13.27618261546587, −12.95737475246501, −12.49606595687410, −11.66166503901248, −11.29627968369998, −10.96470751127013, −10.35450216097851, −9.731070184151957, −9.074757571468057, −8.525284333397267, −7.882658439244292, −7.530289326869333, −6.574497714714213, −6.260828412422607, −5.647527473239625, −4.991979870803420, −4.303561966747623, −3.683787250434934, −3.146343836010946, −2.183793373908613, −1.184303918518846, −0.6734638275894825, 0.6734638275894825, 1.184303918518846, 2.183793373908613, 3.146343836010946, 3.683787250434934, 4.303561966747623, 4.991979870803420, 5.647527473239625, 6.260828412422607, 6.574497714714213, 7.530289326869333, 7.882658439244292, 8.525284333397267, 9.074757571468057, 9.731070184151957, 10.35450216097851, 10.96470751127013, 11.29627968369998, 11.66166503901248, 12.49606595687410, 12.95737475246501, 13.27618261546587, 13.98565083309624, 14.57552609496652, 15.09643309124633

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