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Properties

Label 2-13860-1.1-c1-0-12
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 + 7·17-s + 3·19-s − 3·23-s + 25-s + 5·29-s + 35-s + 2·37-s + 43-s − 8·47-s + 49-s + 9·53-s + 55-s + 9·59-s + 5·61-s + 2·65-s − 2·67-s + 12·71-s + 77-s − 14·79-s − 9·83-s + 7·85-s − 5·89-s + 2·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s + 0.301·11-s + 0.554·13-s + 1.69·17-s + 0.688·19-s − 0.625·23-s + 1/5·25-s + 0.928·29-s + 0.169·35-s + 0.328·37-s + 0.152·43-s − 1.16·47-s + 1/7·49-s + 1.23·53-s + 0.134·55-s + 1.17·59-s + 0.640·61-s + 0.248·65-s − 0.244·67-s + 1.42·71-s + 0.113·77-s − 1.57·79-s − 0.987·83-s + 0.759·85-s − 0.529·89-s + 0.209·91-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 13860,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.166487540\)
\(L(\frac12)\) \(\approx\) \(3.166487540\)
\(L(\frac{3}{2})\) 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 \( 1 - 7 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 5 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 9 T + p T^{2} \)
89 \( 1 + 5 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−16.19309555618611, −15.74880606386578, −14.85905648091890, −14.47086750831683, −13.97175313743307, −13.49323808758464, −12.75580987537062, −12.18932605140210, −11.63876273393800, −11.16033664494508, −10.23247792131017, −9.976107752300868, −9.374379435637291, −8.456733689993211, −8.190483672438686, −7.364087865901059, −6.777102444537223, −5.918716334130784, −5.547964529521463, −4.815652001972896, −3.948745593383849, −3.309260483109833, −2.490872414321260, −1.495053799853079, −0.8724783165281182, 0.8724783165281182, 1.495053799853079, 2.490872414321260, 3.309260483109833, 3.948745593383849, 4.815652001972896, 5.547964529521463, 5.918716334130784, 6.777102444537223, 7.364087865901059, 8.190483672438686, 8.456733689993211, 9.374379435637291, 9.976107752300868, 10.23247792131017, 11.16033664494508, 11.63876273393800, 12.18932605140210, 12.75580987537062, 13.49323808758464, 13.97175313743307, 14.47086750831683, 14.85905648091890, 15.74880606386578, 16.19309555618611

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