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Properties

Label 2-13860-1.1-c1-0-4
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 − 2·17-s − 6·19-s + 4·23-s + 25-s + 4·29-s + 35-s + 6·37-s − 8·43-s + 4·47-s + 49-s + 6·53-s − 55-s − 4·59-s + 6·61-s − 4·65-s − 4·67-s − 77-s − 6·79-s − 14·83-s + 2·85-s + 2·89-s − 4·91-s + 6·95-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 + 0.834·23-s + 1/5·25-s + 0.742·29-s + 0.169·35-s + 0.986·37-s − 1.21·43-s + 0.583·47-s + 1/7·49-s + 0.824·53-s − 0.134·55-s − 0.520·59-s + 0.768·61-s − 0.496·65-s − 0.488·67-s − 0.113·77-s − 0.675·79-s − 1.53·83-s + 0.216·85-s + 0.211·89-s − 0.419·91-s + 0.615·95-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: \(0\)
Selberg data: \((2,\ 13860,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.692312944\)
\(L(\frac12)\) \(\approx\) \(1.692312944\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−16.15423285900462, −15.59749541626428, −15.05381716516045, −14.68024365345238, −13.78174992983666, −13.37604706154611, −12.78351547928875, −12.32575754750369, −11.44952092617426, −11.18120892202529, −10.49103010275493, −9.951488768887052, −9.074632292761225, −8.625451102322668, −8.217090800872673, −7.286216588948450, −6.677448961217597, −6.236554894164155, −5.491476222841481, −4.503196857966845, −4.113500194792253, −3.304341315358302, −2.581763411232414, −1.565806819007471, −0.5912234356529526, 0.5912234356529526, 1.565806819007471, 2.581763411232414, 3.304341315358302, 4.113500194792253, 4.503196857966845, 5.491476222841481, 6.236554894164155, 6.677448961217597, 7.286216588948450, 8.217090800872673, 8.625451102322668, 9.074632292761225, 9.951488768887052, 10.49103010275493, 11.18120892202529, 11.44952092617426, 12.32575754750369, 12.78351547928875, 13.37604706154611, 13.78174992983666, 14.68024365345238, 15.05381716516045, 15.59749541626428, 16.15423285900462

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