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Properties

Label 2-13860-1.1-c1-0-9
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 + 6·17-s − 4·19-s + 25-s + 6·29-s + 8·31-s + 35-s − 10·37-s − 6·41-s + 8·43-s + 49-s + 6·53-s − 55-s − 6·59-s − 4·61-s + 2·65-s + 14·67-s + 2·73-s − 77-s − 10·79-s + 6·83-s + 6·85-s + 18·89-s + 2·91-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 0.301·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s + 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.169·35-s − 1.64·37-s − 0.937·41-s + 1.21·43-s + 1/7·49-s + 0.824·53-s − 0.134·55-s − 0.781·59-s − 0.512·61-s + 0.248·65-s + 1.71·67-s + 0.234·73-s − 0.113·77-s − 1.12·79-s + 0.658·83-s + 0.650·85-s + 1.90·89-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

\(L(1)\) \(\approx\) \(2.747317148\)
\(L(\frac12)\) \(\approx\) \(2.747317148\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 18 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−16.06224791076905, −15.67048510073187, −15.05707215392168, −14.36283168266046, −13.96964789474767, −13.47677659891147, −12.78419112774413, −12.08126119832131, −11.88410567299841, −10.82454826888943, −10.48415372973263, −10.00045340681437, −9.255653008188704, −8.473687068430590, −8.203742590418562, −7.420573233741220, −6.636712005906285, −6.124661738261961, −5.375647184131017, −4.854464653825177, −4.015615409729303, −3.238394910518596, −2.481438929792851, −1.594094663162360, −0.7711488750876141, 0.7711488750876141, 1.594094663162360, 2.481438929792851, 3.238394910518596, 4.015615409729303, 4.854464653825177, 5.375647184131017, 6.124661738261961, 6.636712005906285, 7.420573233741220, 8.203742590418562, 8.473687068430590, 9.255653008188704, 10.00045340681437, 10.48415372973263, 10.82454826888943, 11.88410567299841, 12.08126119832131, 12.78419112774413, 13.47677659891147, 13.96964789474767, 14.36283168266046, 15.05707215392168, 15.67048510073187, 16.06224791076905

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