Properties

Label 2-32340-1.1-c1-0-23
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 − 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: $\chi_{32340} (1, \cdot )$
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 \( 1 + 5 T + p T^{2} \)
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

−15.24467614037508, −14.78091850524857, −14.18258205280799, −13.92307700784939, −12.95427722205499, −12.63748205479164, −12.29385987352669, −11.48538734004364, −11.15557723994746, −10.49544305333315, −9.883712699067078, −9.551169074560142, −8.999081763354955, −8.134809321111673, −7.666904744003992, −7.001753527606593, −6.437919797908044, −5.962398515004928, −5.186549731562485, −4.819235478875638, −4.085748789773449, −3.369090284716143, −2.378906019103701, −1.977535909080888, −0.9262514379383292, 0, 0.9262514379383292, 1.977535909080888, 2.378906019103701, 3.369090284716143, 4.085748789773449, 4.819235478875638, 5.186549731562485, 5.962398515004928, 6.437919797908044, 7.001753527606593, 7.666904744003992, 8.134809321111673, 8.999081763354955, 9.551169074560142, 9.883712699067078, 10.49544305333315, 11.15557723994746, 11.48538734004364, 12.29385987352669, 12.63748205479164, 12.95427722205499, 13.92307700784939, 14.18258205280799, 14.78091850524857, 15.24467614037508

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