Properties

Label 2-13872-1.1-c1-0-34
Degree $2$
Conductor $13872$
Sign $-1$
Analytic cond. $110.768$
Root an. cond. $10.5246$
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 + 2·5-s + 9-s + 4·11-s − 2·13-s − 2·15-s + 4·19-s − 8·23-s − 25-s − 27-s − 6·29-s + 8·31-s − 4·33-s − 6·37-s + 2·39-s + 6·41-s − 4·43-s + 2·45-s − 7·49-s − 2·53-s + 8·55-s − 4·57-s − 4·59-s + 2·61-s − 4·65-s + 4·67-s + 8·69-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s − 0.554·13-s − 0.516·15-s + 0.917·19-s − 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.696·33-s − 0.986·37-s + 0.320·39-s + 0.937·41-s − 0.609·43-s + 0.298·45-s − 49-s − 0.274·53-s + 1.07·55-s − 0.529·57-s − 0.520·59-s + 0.256·61-s − 0.496·65-s + 0.488·67-s + 0.963·69-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13872\)    =    \(2^{4} \cdot 3 \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(110.768\)
Root analytic conductor: \(10.5246\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{13872} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 13872,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
17 \( 1 \)
good5 \( 1 - 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−16.49087773919892, −15.99633748301758, −15.41845636346975, −14.60044054245105, −14.18371932720275, −13.70668233877805, −13.14951520798310, −12.29966189046609, −11.95890114392885, −11.48056561580270, −10.73781930283072, −9.984088692506675, −9.646830925040174, −9.248365896146778, −8.259387310160539, −7.703185711167563, −6.819318573578103, −6.432747659713858, −5.710596573061130, −5.300281795275805, −4.360132705504210, −3.813057485513455, −2.811774165493996, −1.864017865141919, −1.298455202621086, 0, 1.298455202621086, 1.864017865141919, 2.811774165493996, 3.813057485513455, 4.360132705504210, 5.300281795275805, 5.710596573061130, 6.432747659713858, 6.819318573578103, 7.703185711167563, 8.259387310160539, 9.248365896146778, 9.646830925040174, 9.984088692506675, 10.73781930283072, 11.48056561580270, 11.95890114392885, 12.29966189046609, 13.14951520798310, 13.70668233877805, 14.18371932720275, 14.60044054245105, 15.41845636346975, 15.99633748301758, 16.49087773919892

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