Properties

Label 2-14586-1.1-c1-0-5
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 − 6-s − 4·7-s − 8-s + 9-s + 11-s + 12-s + 13-s + 4·14-s + 16-s + 17-s − 18-s + 2·19-s − 4·21-s − 22-s − 24-s − 5·25-s − 26-s + 27-s − 4·28-s − 6·29-s + 8·31-s − 32-s + 33-s − 34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s + 0.301·11-s + 0.288·12-s + 0.277·13-s + 1.06·14-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.458·19-s − 0.872·21-s − 0.213·22-s − 0.204·24-s − 25-s − 0.196·26-s + 0.192·27-s − 0.755·28-s − 1.11·29-s + 1.43·31-s − 0.176·32-s + 0.174·33-s − 0.171·34-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 \( 1 + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
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

−16.35525550660073, −15.91246009132617, −15.31458640639723, −14.95112873429007, −14.04522686710640, −13.55575568073967, −13.08557863710958, −12.45844574052529, −11.79466565758597, −11.35285952220832, −10.32623453338424, −10.02231301300582, −9.514315663408493, −8.996583020867584, −8.407701791190411, −7.645874030580592, −7.203648107263886, −6.350627569344427, −6.105213602338712, −5.136982400790281, −4.040148490103989, −3.474817056153517, −2.885289530296387, −2.056866367740717, −1.070129089468315, 0, 1.070129089468315, 2.056866367740717, 2.885289530296387, 3.474817056153517, 4.040148490103989, 5.136982400790281, 6.105213602338712, 6.350627569344427, 7.203648107263886, 7.645874030580592, 8.407701791190411, 8.996583020867584, 9.514315663408493, 10.02231301300582, 10.32623453338424, 11.35285952220832, 11.79466565758597, 12.45844574052529, 13.08557863710958, 13.55575568073967, 14.04522686710640, 14.95112873429007, 15.31458640639723, 15.91246009132617, 16.35525550660073

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