Properties

Label 2-13860-1.1-c1-0-5
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 + 2·17-s + 4·19-s + 6·23-s + 25-s + 10·29-s − 2·31-s − 35-s + 2·37-s − 8·41-s + 12·47-s + 49-s − 2·53-s + 55-s + 10·59-s − 4·61-s + 2·65-s − 10·67-s − 8·71-s − 14·73-s − 77-s − 4·83-s − 2·85-s + 6·89-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.301·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s + 1.25·23-s + 1/5·25-s + 1.85·29-s − 0.359·31-s − 0.169·35-s + 0.328·37-s − 1.24·41-s + 1.75·47-s + 1/7·49-s − 0.274·53-s + 0.134·55-s + 1.30·59-s − 0.512·61-s + 0.248·65-s − 1.22·67-s − 0.949·71-s − 1.63·73-s − 0.113·77-s − 0.439·83-s − 0.216·85-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: $\chi_{13860} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 13860,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.067640152\)
\(L(\frac12)\) \(\approx\) \(2.067640152\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 - 10 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
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

−16.15115712303675, −15.61522594403560, −15.01291985964961, −14.58518846347397, −13.93575428506549, −13.41582357750965, −12.75360564293400, −11.99454192121687, −11.85686023508517, −11.06949283204286, −10.38501239478919, −10.03389084863462, −9.130215093507948, −8.666466589418814, −7.973619983898125, −7.336711404398971, −6.985905960322529, −6.031347881423650, −5.282857065777569, −4.801236436077981, −4.101739389049672, −3.103244991137182, −2.716884055039908, −1.504125346386899, −0.6731977625801532, 0.6731977625801532, 1.504125346386899, 2.716884055039908, 3.103244991137182, 4.101739389049672, 4.801236436077981, 5.282857065777569, 6.031347881423650, 6.985905960322529, 7.336711404398971, 7.973619983898125, 8.666466589418814, 9.130215093507948, 10.03389084863462, 10.38501239478919, 11.06949283204286, 11.85686023508517, 11.99454192121687, 12.75360564293400, 13.41582357750965, 13.93575428506549, 14.58518846347397, 15.01291985964961, 15.61522594403560, 16.15115712303675

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