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Properties

Label 2-13860-1.1-c1-0-14
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 + 2·13-s + 6·17-s + 8·19-s + 6·23-s + 25-s − 6·29-s + 2·31-s + 35-s + 2·37-s + 8·43-s + 12·47-s + 49-s − 6·53-s + 55-s − 6·59-s + 8·61-s + 2·65-s + 2·67-s − 10·73-s + 77-s + 8·79-s − 12·83-s + 6·85-s − 6·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 0.301·11-s + 0.554·13-s + 1.45·17-s + 1.83·19-s + 1.25·23-s + 1/5·25-s − 1.11·29-s + 0.359·31-s + 0.169·35-s + 0.328·37-s + 1.21·43-s + 1.75·47-s + 1/7·49-s − 0.824·53-s + 0.134·55-s − 0.781·59-s + 1.02·61-s + 0.248·65-s + 0.244·67-s − 1.17·73-s + 0.113·77-s + 0.900·79-s − 1.31·83-s + 0.650·85-s − 0.635·89-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: \(0\)
Selberg data: \((2,\ 13860,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.450335498\)
\(L(\frac12)\) \(\approx\) \(3.450335498\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−16.15982327553957, −15.66071911456155, −14.98806024310024, −14.39368639511253, −13.95269770812627, −13.52700909278121, −12.68937498666040, −12.32871167869986, −11.44176821095888, −11.24659370708362, −10.40030769975272, −9.834159692292568, −9.225376316091283, −8.831675513718556, −7.813169929545107, −7.514580439704463, −6.825358802554139, −5.816480295964995, −5.579311701022029, −4.864690981912891, −3.934199664608699, −3.258531472399650, −2.575234954357342, −1.369533771063003, −0.9750830777151739, 0.9750830777151739, 1.369533771063003, 2.575234954357342, 3.258531472399650, 3.934199664608699, 4.864690981912891, 5.579311701022029, 5.816480295964995, 6.825358802554139, 7.514580439704463, 7.813169929545107, 8.831675513718556, 9.225376316091283, 9.834159692292568, 10.40030769975272, 11.24659370708362, 11.44176821095888, 12.32871167869986, 12.68937498666040, 13.52700909278121, 13.95269770812627, 14.39368639511253, 14.98806024310024, 15.66071911456155, 16.15982327553957

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