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Properties

Label 2-13860-1.1-c1-0-22
Degree $2$
Conductor $13860$
Sign $-1$
Analytic cond. $110.672$
Root an. cond. $10.5201$
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  + 5-s + 7-s + 11-s − 4·13-s − 2·17-s − 6·19-s + 6·23-s + 25-s − 4·29-s + 6·31-s + 35-s + 2·37-s − 6·41-s + 4·43-s − 8·47-s + 49-s + 6·53-s + 55-s − 12·59-s + 14·61-s − 4·65-s + 4·67-s − 12·73-s + 77-s + 4·79-s − 2·85-s − 2·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 0.301·11-s − 1.10·13-s − 0.485·17-s − 1.37·19-s + 1.25·23-s + 1/5·25-s − 0.742·29-s + 1.07·31-s + 0.169·35-s + 0.328·37-s − 0.937·41-s + 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.824·53-s + 0.134·55-s − 1.56·59-s + 1.79·61-s − 0.496·65-s + 0.488·67-s − 1.40·73-s + 0.113·77-s + 0.450·79-s − 0.216·85-s − 0.211·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 13860 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 13860 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(13860\)    =    \(2^{2} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(110.672\)
Root analytic conductor: \(10.5201\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 13860,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 - T \)
7 \( 1 - T \)
11 \( 1 - T \)
good13 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 14 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 12 T + p T^{2} \)
79 \( 1 - 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−16.68995345763163, −15.85033872654638, −15.03186302159668, −14.91556551569507, −14.32790943894046, −13.57548234039221, −13.08935561044716, −12.58357277446698, −11.92491892867969, −11.29913728100017, −10.79200934791645, −10.11285603420050, −9.611716137529898, −8.893972063700191, −8.461983144417931, −7.680242383635702, −6.967404714548047, −6.518778904568047, −5.759567308122782, −4.927646440174265, −4.580199810378687, −3.695119953054442, −2.689033116450664, −2.159545255819726, −1.232282806233401, 0, 1.232282806233401, 2.159545255819726, 2.689033116450664, 3.695119953054442, 4.580199810378687, 4.927646440174265, 5.759567308122782, 6.518778904568047, 6.967404714548047, 7.680242383635702, 8.461983144417931, 8.893972063700191, 9.611716137529898, 10.11285603420050, 10.79200934791645, 11.29913728100017, 11.92491892867969, 12.58357277446698, 13.08935561044716, 13.57548234039221, 14.32790943894046, 14.91556551569507, 15.03186302159668, 15.85033872654638, 16.68995345763163

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