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Properties

Label 2-13860-1.1-c1-0-6
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 + 4·13-s + 5·17-s − 5·19-s + 3·23-s + 25-s + 29-s − 2·31-s − 35-s − 4·37-s + 10·41-s − 9·43-s − 6·47-s + 49-s + 7·53-s + 55-s + 59-s + 5·61-s − 4·65-s + 2·67-s + 10·71-s + 4·73-s − 77-s + 5·83-s − 5·85-s + ⋯
L(s)  = 1  − 0.447·5-s + 0.377·7-s − 0.301·11-s + 1.10·13-s + 1.21·17-s − 1.14·19-s + 0.625·23-s + 1/5·25-s + 0.185·29-s − 0.359·31-s − 0.169·35-s − 0.657·37-s + 1.56·41-s − 1.37·43-s − 0.875·47-s + 1/7·49-s + 0.961·53-s + 0.134·55-s + 0.130·59-s + 0.640·61-s − 0.496·65-s + 0.244·67-s + 1.18·71-s + 0.468·73-s − 0.113·77-s + 0.548·83-s − 0.542·85-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

\(L(1)\) \(\approx\) \(2.115652481\)
\(L(\frac12)\) \(\approx\) \(2.115652481\)
\(L(\frac{3}{2})\) 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 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 5 T + p T^{2} \)
19 \( 1 + 5 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 7 T + p T^{2} \)
59 \( 1 - T + p T^{2} \)
61 \( 1 - 5 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 5 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−16.26243008865179, −15.44708649076953, −15.21463896620956, −14.37271330205174, −14.15414694405908, −13.15154388375515, −12.92441392215056, −12.21024355129564, −11.56512897689502, −11.06997615955655, −10.53677080338364, −9.976496718659717, −9.139994114237897, −8.506388128258873, −8.128117962205010, −7.463401441852466, −6.743032539673490, −6.108034588405528, −5.364661143393378, −4.785044033533333, −3.876007255469492, −3.453145052558960, −2.496403092207290, −1.556163619965366, −0.6744760419086420, 0.6744760419086420, 1.556163619965366, 2.496403092207290, 3.453145052558960, 3.876007255469492, 4.785044033533333, 5.364661143393378, 6.108034588405528, 6.743032539673490, 7.463401441852466, 8.128117962205010, 8.506388128258873, 9.139994114237897, 9.976496718659717, 10.53677080338364, 11.06997615955655, 11.56512897689502, 12.21024355129564, 12.92441392215056, 13.15154388375515, 14.15414694405908, 14.37271330205174, 15.21463896620956, 15.44708649076953, 16.26243008865179

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