Properties

Label 2-14586-1.1-c1-0-8
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 + 2·5-s − 6-s + 7-s − 8-s + 9-s − 2·10-s + 11-s + 12-s − 13-s − 14-s + 2·15-s + 16-s − 17-s − 18-s − 4·19-s + 2·20-s + 21-s − 22-s − 6·23-s − 24-s − 25-s + 26-s + 27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.894·5-s − 0.408·6-s + 0.377·7-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 − 0.267·14-s + 0.516·15-s + 1/4·16-s − 0.242·17-s − 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.218·21-s − 0.213·22-s − 1.25·23-s − 0.204·24-s − 1/5·25-s + 0.196·26-s + 0.192·27-s + 0.188·28-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: Trivial
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 \( 1 - 2 T + p T^{2} \)
7 \( 1 - T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 7 T + p T^{2} \)
43 \( 1 - 3 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 7 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 7 T + p T^{2} \)
89 \( 1 + 3 T + p T^{2} \)
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

−16.45513863218395, −15.86542920284487, −15.17669314055710, −14.76746638734714, −14.19978551636194, −13.54183588003769, −13.25158442036799, −12.28380043768303, −11.94563405264425, −11.13764057589382, −10.46883989397217, −10.06347690871370, −9.462299488794444, −8.947687459126287, −8.373047191561505, −7.805227960114998, −7.148183830551460, −6.420137235206918, −5.912845580357937, −5.136636276652032, −4.236400694474854, −3.588574780045693, −2.469132827969466, −2.059695776447628, −1.361902187761392, 0, 1.361902187761392, 2.059695776447628, 2.469132827969466, 3.588574780045693, 4.236400694474854, 5.136636276652032, 5.912845580357937, 6.420137235206918, 7.148183830551460, 7.805227960114998, 8.373047191561505, 8.947687459126287, 9.462299488794444, 10.06347690871370, 10.46883989397217, 11.13764057589382, 11.94563405264425, 12.28380043768303, 13.25158442036799, 13.54183588003769, 14.19978551636194, 14.76746638734714, 15.17669314055710, 15.86542920284487, 16.45513863218395

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