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Properties

Label 2-13860-1.1-c1-0-7
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 + 2·23-s + 25-s + 2·29-s − 10·31-s + 35-s + 2·37-s + 12·41-s + 49-s + 6·53-s − 55-s − 6·59-s − 2·65-s + 2·67-s + 8·71-s − 10·73-s − 77-s − 4·83-s − 6·85-s + 6·89-s − 2·91-s − 4·95-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 + 0.417·23-s + 1/5·25-s + 0.371·29-s − 1.79·31-s + 0.169·35-s + 0.328·37-s + 1.87·41-s + 1/7·49-s + 0.824·53-s − 0.134·55-s − 0.781·59-s − 0.248·65-s + 0.244·67-s + 0.949·71-s − 1.17·73-s − 0.113·77-s − 0.439·83-s − 0.650·85-s + 0.635·89-s − 0.209·91-s − 0.410·95-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

\(L(1)\) \(\approx\) \(2.154380694\)
\(L(\frac12)\) \(\approx\) \(2.154380694\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
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

−16.07204971283662, −15.85169607810995, −14.92242210006567, −14.55129907492257, −14.01086502894185, −13.33008530018026, −12.71269200900230, −12.24763235699079, −11.68089057124068, −11.02248405520145, −10.58104120102881, −9.701966949808325, −9.357592035952480, −8.674062656954330, −7.911932319851381, −7.415905253608811, −6.886410118343299, −5.899290125774788, −5.606415346088058, −4.707874916102609, −3.840713999224576, −3.397253984807483, −2.645986192446161, −1.448802054030251, −0.7059624685041476, 0.7059624685041476, 1.448802054030251, 2.645986192446161, 3.397253984807483, 3.840713999224576, 4.707874916102609, 5.606415346088058, 5.899290125774788, 6.886410118343299, 7.415905253608811, 7.911932319851381, 8.674062656954330, 9.357592035952480, 9.701966949808325, 10.58104120102881, 11.02248405520145, 11.68089057124068, 12.24763235699079, 12.71269200900230, 13.33008530018026, 14.01086502894185, 14.55129907492257, 14.92242210006567, 15.85169607810995, 16.07204971283662

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