Properties

Label 2-13860-1.1-c1-0-10
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 + 6·13-s + 2·17-s + 8·19-s − 8·23-s + 25-s + 6·29-s − 35-s − 2·37-s − 6·41-s + 8·43-s + 8·47-s + 49-s + 2·53-s − 55-s − 6·59-s − 8·61-s − 6·65-s + 10·67-s + 8·71-s + 14·73-s + 77-s + 10·79-s − 18·83-s − 2·85-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s + 0.301·11-s + 1.66·13-s + 0.485·17-s + 1.83·19-s − 1.66·23-s + 1/5·25-s + 1.11·29-s − 0.169·35-s − 0.328·37-s − 0.937·41-s + 1.21·43-s + 1.16·47-s + 1/7·49-s + 0.274·53-s − 0.134·55-s − 0.781·59-s − 1.02·61-s − 0.744·65-s + 1.22·67-s + 0.949·71-s + 1.63·73-s + 0.113·77-s + 1.12·79-s − 1.97·83-s − 0.216·85-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

\(L(1)\) \(\approx\) \(2.667411026\)
\(L(\frac12)\) \(\approx\) \(2.667411026\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 18 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + p T^{2} \)
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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

−16.09821756310367, −15.57858405896052, −15.34854226311797, −14.13583000552580, −13.98933412141738, −13.69905483504023, −12.58348332240626, −12.21474004126537, −11.57455129241971, −11.21778416423966, −10.42882141446991, −9.966575687830797, −9.129678143673020, −8.654733641678397, −7.857431531589266, −7.671659856190925, −6.667258792134139, −6.096293691204265, −5.454995072393265, −4.735771187158683, −3.742365192764687, −3.597467577939297, −2.516406067405897, −1.433496301360454, −0.8105975320587855, 0.8105975320587855, 1.433496301360454, 2.516406067405897, 3.597467577939297, 3.742365192764687, 4.735771187158683, 5.454995072393265, 6.096293691204265, 6.667258792134139, 7.671659856190925, 7.857431531589266, 8.654733641678397, 9.129678143673020, 9.966575687830797, 10.42882141446991, 11.21778416423966, 11.57455129241971, 12.21474004126537, 12.58348332240626, 13.69905483504023, 13.98933412141738, 14.13583000552580, 15.34854226311797, 15.57858405896052, 16.09821756310367

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