Properties

Label 2-32340-1.1-c1-0-32
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 − 2·13-s − 15-s + 3·17-s + 7·19-s − 3·23-s + 25-s − 27-s − 3·29-s + 4·31-s − 33-s − 10·37-s + 2·39-s − 7·43-s + 45-s − 3·51-s + 9·53-s + 55-s − 7·57-s − 3·59-s + 61-s − 2·65-s + 2·67-s + 3·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.447·5-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.258·15-s + 0.727·17-s + 1.60·19-s − 0.625·23-s + 1/5·25-s − 0.192·27-s − 0.557·29-s + 0.718·31-s − 0.174·33-s − 1.64·37-s + 0.320·39-s − 1.06·43-s + 0.149·45-s − 0.420·51-s + 1.23·53-s + 0.134·55-s − 0.927·57-s − 0.390·59-s + 0.128·61-s − 0.248·65-s + 0.244·67-s + 0.361·69-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: $\chi_{32340} (1, \cdot )$
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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 7 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 19 T + p T^{2} \)
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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

−15.33312228630847, −14.75641519868434, −14.20419680966124, −13.67929151050631, −13.34982213269468, −12.48932629840199, −11.99959804641611, −11.82233407107813, −11.09829874101964, −10.36619517347577, −9.990207121777554, −9.573868227406260, −8.921271245346757, −8.226600796838364, −7.543464538844172, −7.093698585117175, −6.486990203992719, −5.779283107559642, −5.303183423763698, −4.888344107769409, −3.957763310624098, −3.366307821576286, −2.603982626589774, −1.687202216611338, −1.080339058929199, 0, 1.080339058929199, 1.687202216611338, 2.603982626589774, 3.366307821576286, 3.957763310624098, 4.888344107769409, 5.303183423763698, 5.779283107559642, 6.486990203992719, 7.093698585117175, 7.543464538844172, 8.226600796838364, 8.921271245346757, 9.573868227406260, 9.990207121777554, 10.36619517347577, 11.09829874101964, 11.82233407107813, 11.99959804641611, 12.48932629840199, 13.34982213269468, 13.67929151050631, 14.20419680966124, 14.75641519868434, 15.33312228630847

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