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Properties

Label 2-13860-1.1-c1-0-13
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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 13860,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.160785135\)
\(L(\frac12)\) \(\approx\) \(3.160785135\)
\(L(\frac{3}{2})\) 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

−16.20144246267293, −15.63010798607722, −15.10078925479961, −14.45034494588015, −13.78288438322335, −13.45016750482752, −12.94115681256921, −12.26473408836257, −11.43859779354032, −10.89960575574924, −10.83614973227915, −9.617915910101855, −9.351379930022327, −8.707849376933874, −8.016792636489767, −7.407415763534709, −6.736792178834556, −6.033346625833961, −5.372482040212457, −4.937851832953631, −3.900627776322809, −3.313972306021135, −2.507809402414670, −1.484432197965793, −0.8745078422650734, 0.8745078422650734, 1.484432197965793, 2.507809402414670, 3.313972306021135, 3.900627776322809, 4.937851832953631, 5.372482040212457, 6.033346625833961, 6.736792178834556, 7.407415763534709, 8.016792636489767, 8.707849376933874, 9.351379930022327, 9.617915910101855, 10.83614973227915, 10.89960575574924, 11.43859779354032, 12.26473408836257, 12.94115681256921, 13.45016750482752, 13.78288438322335, 14.45034494588015, 15.10078925479961, 15.63010798607722, 16.20144246267293

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