Properties

Label 2-32340-1.1-c1-0-1
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 − 2·19-s + 4·23-s + 25-s + 27-s − 4·29-s − 8·31-s − 33-s − 10·37-s − 4·39-s − 8·43-s − 45-s − 12·47-s − 2·51-s + 10·53-s + 55-s − 2·57-s − 12·59-s + 2·61-s + 4·65-s − 12·67-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 − 0.458·19-s + 0.834·23-s + 1/5·25-s + 0.192·27-s − 0.742·29-s − 1.43·31-s − 0.174·33-s − 1.64·37-s − 0.640·39-s − 1.21·43-s − 0.149·45-s − 1.75·47-s − 0.280·51-s + 1.37·53-s + 0.134·55-s − 0.264·57-s − 1.56·59-s + 0.256·61-s + 0.496·65-s − 1.46·67-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

\(L(1)\) \(\approx\) \(1.103808455\)
\(L(\frac12)\) \(\approx\) \(1.103808455\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
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

−15.05549508016645, −14.81290287184401, −13.90754134908992, −13.54495440298157, −12.87643683566152, −12.50780406537536, −11.95363613488216, −11.27322463250825, −10.77823324637207, −10.23627688015417, −9.607206292528585, −9.011789642803549, −8.659142792963247, −7.872556708761042, −7.484513438622666, −6.893794871030726, −6.414444913106237, −5.288278256307785, −5.042180059634993, −4.295108404747525, −3.508613944638171, −3.108587606347548, −2.152263010947926, −1.716172209051663, −0.3617594754044390, 0.3617594754044390, 1.716172209051663, 2.152263010947926, 3.108587606347548, 3.508613944638171, 4.295108404747525, 5.042180059634993, 5.288278256307785, 6.414444913106237, 6.893794871030726, 7.484513438622666, 7.872556708761042, 8.659142792963247, 9.011789642803549, 9.607206292528585, 10.23627688015417, 10.77823324637207, 11.27322463250825, 11.95363613488216, 12.50780406537536, 12.87643683566152, 13.54495440298157, 13.90754134908992, 14.81290287184401, 15.05549508016645

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