Properties

Label 2-32340-1.1-c1-0-38
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 − 2·17-s + 6·19-s + 25-s − 27-s + 2·29-s − 4·31-s − 33-s − 2·37-s − 2·39-s + 4·43-s + 45-s − 8·47-s + 2·51-s − 4·53-s + 55-s − 6·57-s + 14·59-s − 14·61-s + 2·65-s − 2·67-s − 8·71-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 − 0.485·17-s + 1.37·19-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.174·33-s − 0.328·37-s − 0.320·39-s + 0.609·43-s + 0.149·45-s − 1.16·47-s + 0.280·51-s − 0.549·53-s + 0.134·55-s − 0.794·57-s + 1.82·59-s − 1.79·61-s + 0.248·65-s − 0.244·67-s − 0.949·71-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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−15.41447872341532, −14.75143638721531, −14.13845838971472, −13.79028457866020, −13.12012652772544, −12.76015610211224, −12.03930189217895, −11.57536145208735, −11.10582275997894, −10.56653492320684, −9.930554973477150, −9.479900720186197, −8.904596173733009, −8.318604506592693, −7.550391466891519, −7.031912749794534, −6.447564419147635, −5.865243421858637, −5.359265710174193, −4.747681279255700, −4.036528394879325, −3.338453409269666, −2.624732751405673, −1.634761011908527, −1.124998346684711, 0, 1.124998346684711, 1.634761011908527, 2.624732751405673, 3.338453409269666, 4.036528394879325, 4.747681279255700, 5.359265710174193, 5.865243421858637, 6.447564419147635, 7.031912749794534, 7.550391466891519, 8.318604506592693, 8.904596173733009, 9.479900720186197, 9.930554973477150, 10.56653492320684, 11.10582275997894, 11.57536145208735, 12.03930189217895, 12.76015610211224, 13.12012652772544, 13.79028457866020, 14.13845838971472, 14.75143638721531, 15.41447872341532

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