Properties

Label 2-13860-1.1-c1-0-1
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 5-s + 7-s − 11-s − 4·13-s − 4·19-s − 6·23-s + 25-s + 6·29-s − 4·31-s − 35-s + 2·37-s + 6·41-s − 4·43-s − 6·47-s + 49-s + 6·53-s + 55-s − 10·61-s + 4·65-s + 2·67-s − 12·71-s + 8·73-s − 77-s − 16·79-s + 12·83-s + 18·89-s − 4·91-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.301·11-s − 1.10·13-s − 0.917·19-s − 1.25·23-s + 1/5·25-s + 1.11·29-s − 0.718·31-s − 0.169·35-s + 0.328·37-s + 0.937·41-s − 0.609·43-s − 0.875·47-s + 1/7·49-s + 0.824·53-s + 0.134·55-s − 1.28·61-s + 0.496·65-s + 0.244·67-s − 1.42·71-s + 0.936·73-s − 0.113·77-s − 1.80·79-s + 1.31·83-s + 1.90·89-s − 0.419·91-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: $\chi_{13860} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 13860,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.139059472\)
\(L(\frac12)\) \(\approx\) \(1.139059472\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−16.20006312623454, −15.61763500271349, −14.88989023977110, −14.66375546165292, −14.01620246388893, −13.33886389416991, −12.71551080463246, −12.12799290615055, −11.79800176503626, −11.01009674420843, −10.46449070667061, −9.950984896357124, −9.251287599884192, −8.533849107124310, −7.974925438012989, −7.514878197790038, −6.788927274247282, −6.106677234531384, −5.367492993460502, −4.588902217611892, −4.222377149512782, −3.242419301923330, −2.447148372370221, −1.748513522362626, −0.4561081983971308, 0.4561081983971308, 1.748513522362626, 2.447148372370221, 3.242419301923330, 4.222377149512782, 4.588902217611892, 5.367492993460502, 6.106677234531384, 6.788927274247282, 7.514878197790038, 7.974925438012989, 8.533849107124310, 9.251287599884192, 9.950984896357124, 10.46449070667061, 11.01009674420843, 11.79800176503626, 12.12799290615055, 12.71551080463246, 13.33886389416991, 14.01620246388893, 14.66375546165292, 14.88989023977110, 15.61763500271349, 16.20006312623454

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