Properties

Label 2-14586-1.1-c1-0-0
Degree $2$
Conductor $14586$
Sign $1$
Analytic cond. $116.469$
Root an. cond. $10.7921$
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  + 2-s − 3-s + 4-s − 2·5-s − 6-s + 8-s + 9-s − 2·10-s + 11-s − 12-s + 13-s + 2·15-s + 16-s − 17-s + 18-s + 8·19-s − 2·20-s + 22-s − 4·23-s − 24-s − 25-s + 26-s − 27-s + 6·29-s + 2·30-s − 4·31-s + 32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.894·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.632·10-s + 0.301·11-s − 0.288·12-s + 0.277·13-s + 0.516·15-s + 1/4·16-s − 0.242·17-s + 0.235·18-s + 1.83·19-s − 0.447·20-s + 0.213·22-s − 0.834·23-s − 0.204·24-s − 1/5·25-s + 0.196·26-s − 0.192·27-s + 1.11·29-s + 0.365·30-s − 0.718·31-s + 0.176·32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14586\)    =    \(2 \cdot 3 \cdot 11 \cdot 13 \cdot 17\)
Sign: $1$
Analytic conductor: \(116.469\)
Root analytic conductor: \(10.7921\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{14586} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 14586,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.303668043\)
\(L(\frac12)\) \(\approx\) \(2.303668043\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
11 \( 1 - T \)
13 \( 1 - T \)
17 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
7 \( 1 + p T^{2} \)
19 \( 1 - 8 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−15.89263527343359, −15.72400527982683, −15.12743662819552, −14.17900147858361, −14.01239324739613, −13.34014799623573, −12.53901534211780, −12.00388759571822, −11.85791684727675, −11.10517542820850, −10.69832079610569, −9.874954567895536, −9.338705940202399, −8.401089183498101, −7.862985956970226, −7.170630739738205, −6.763904703641339, −5.874326071804082, −5.410472558636607, −4.699927905141964, −3.936910767099510, −3.560927908968328, −2.643227960954291, −1.574967149751973, −0.6356293898749369, 0.6356293898749369, 1.574967149751973, 2.643227960954291, 3.560927908968328, 3.936910767099510, 4.699927905141964, 5.410472558636607, 5.874326071804082, 6.763904703641339, 7.170630739738205, 7.862985956970226, 8.401089183498101, 9.338705940202399, 9.874954567895536, 10.69832079610569, 11.10517542820850, 11.85791684727675, 12.00388759571822, 12.53901534211780, 13.34014799623573, 14.01239324739613, 14.17900147858361, 15.12743662819552, 15.72400527982683, 15.89263527343359

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