Properties

Label 2-13872-1.1-c1-0-22
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 − 3·5-s + 9-s − 11-s + 3·13-s + 3·15-s − 19-s + 7·23-s + 4·25-s − 27-s − 6·29-s − 2·31-s + 33-s + 4·37-s − 3·39-s − 9·41-s + 43-s − 3·45-s − 10·47-s − 7·49-s − 2·53-s + 3·55-s + 57-s + 6·59-s + 12·61-s − 9·65-s + 4·67-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.34·5-s + 1/3·9-s − 0.301·11-s + 0.832·13-s + 0.774·15-s − 0.229·19-s + 1.45·23-s + 4/5·25-s − 0.192·27-s − 1.11·29-s − 0.359·31-s + 0.174·33-s + 0.657·37-s − 0.480·39-s − 1.40·41-s + 0.152·43-s − 0.447·45-s − 1.45·47-s − 49-s − 0.274·53-s + 0.404·55-s + 0.132·57-s + 0.781·59-s + 1.53·61-s − 1.11·65-s + 0.488·67-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: Trivial
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 \( 1 + 3 T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 - 7 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−16.33520979276125, −16.01993468730280, −15.26880695654303, −15.00566149180374, −14.39947362650378, −13.33835697565993, −13.06855445723225, −12.52819960233137, −11.68269777759095, −11.38616646035349, −10.98483620757991, −10.34353289631360, −9.542180469758720, −8.890560650363459, −8.149425457605884, −7.829007475797260, −6.920841893350595, −6.645904910459483, −5.649246337786496, −5.059162759674450, −4.395099804755680, −3.616871266840290, −3.203644868826525, −1.951085675697849, −0.9215455229561430, 0, 0.9215455229561430, 1.951085675697849, 3.203644868826525, 3.616871266840290, 4.395099804755680, 5.059162759674450, 5.649246337786496, 6.645904910459483, 6.920841893350595, 7.829007475797260, 8.149425457605884, 8.890560650363459, 9.542180469758720, 10.34353289631360, 10.98483620757991, 11.38616646035349, 11.68269777759095, 12.52819960233137, 13.06855445723225, 13.33835697565993, 14.39947362650378, 15.00566149180374, 15.26880695654303, 16.01993468730280, 16.33520979276125

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