Properties

Label 2-14586-1.1-c1-0-2
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 3-s + 4-s − 4·5-s − 6-s − 2·7-s − 8-s + 9-s + 4·10-s + 11-s + 12-s − 13-s + 2·14-s − 4·15-s + 16-s − 17-s − 18-s − 4·19-s − 4·20-s − 2·21-s − 22-s − 6·23-s − 24-s + 11·25-s + 26-s + 27-s − 2·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 1.78·5-s − 0.408·6-s − 0.755·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.301·11-s + 0.288·12-s − 0.277·13-s + 0.534·14-s − 1.03·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s − 0.894·20-s − 0.436·21-s − 0.213·22-s − 1.25·23-s − 0.204·24-s + 11/5·25-s + 0.196·26-s + 0.192·27-s − 0.377·28-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: \(1\)
Selberg data: \((2,\ 14586,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 - 12 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 2 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 2 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−16.25557131977370, −15.89100102873760, −15.32165057348103, −14.89555472386780, −14.42849754559807, −13.47950573987809, −12.93347820461488, −12.31650766493329, −11.74400315512770, −11.44329036870215, −10.60793604357791, −10.00181516008828, −9.531751868091784, −8.714663329380377, −8.154147057722409, −7.988233761959827, −7.160088345151468, −6.599167278785526, −6.085481112454584, −4.694984931567072, −4.252701369979912, −3.535438749395598, −2.931813966012393, −2.117263684890014, −0.8449661824052050, 0, 0.8449661824052050, 2.117263684890014, 2.931813966012393, 3.535438749395598, 4.252701369979912, 4.694984931567072, 6.085481112454584, 6.599167278785526, 7.160088345151468, 7.988233761959827, 8.154147057722409, 8.714663329380377, 9.531751868091784, 10.00181516008828, 10.60793604357791, 11.44329036870215, 11.74400315512770, 12.31650766493329, 12.93347820461488, 13.47950573987809, 14.42849754559807, 14.89555472386780, 15.32165057348103, 15.89100102873760, 16.25557131977370

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