Properties

Label 2-14586-1.1-c1-0-3
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 + 4·5-s − 6-s − 8-s + 9-s − 4·10-s + 11-s + 12-s − 13-s + 4·15-s + 16-s + 17-s − 18-s + 2·19-s + 4·20-s − 22-s − 4·23-s − 24-s + 11·25-s + 26-s + 27-s + 2·29-s − 4·30-s − 32-s + 33-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.78·5-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.26·10-s + 0.301·11-s + 0.288·12-s − 0.277·13-s + 1.03·15-s + 1/4·16-s + 0.242·17-s − 0.235·18-s + 0.458·19-s + 0.894·20-s − 0.213·22-s − 0.834·23-s − 0.204·24-s + 11/5·25-s + 0.196·26-s + 0.192·27-s + 0.371·29-s − 0.730·30-s − 0.176·32-s + 0.174·33-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

\(L(1)\) \(\approx\) \(3.137035691\)
\(L(\frac12)\) \(\approx\) \(3.137035691\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 10 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 14 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 16 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
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

−16.21090077914068, −15.69896300880333, −14.75114708819012, −14.39757895580699, −14.01024538916436, −13.34435805767725, −12.82206308061949, −12.25026707837201, −11.43831303817082, −10.81908638654190, −9.984590497350799, −9.815891489537064, −9.425168954332504, −8.625303306789774, −8.197513847943483, −7.398282044226276, −6.585007875943309, −6.338691273422551, −5.399185177651725, −4.968615566706756, −3.778117474162491, −3.007741232797420, −2.193082857089771, −1.776527385214073, −0.8578112312733917, 0.8578112312733917, 1.776527385214073, 2.193082857089771, 3.007741232797420, 3.778117474162491, 4.968615566706756, 5.399185177651725, 6.338691273422551, 6.585007875943309, 7.398282044226276, 8.197513847943483, 8.625303306789774, 9.425168954332504, 9.815891489537064, 9.984590497350799, 10.81908638654190, 11.43831303817082, 12.25026707837201, 12.82206308061949, 13.34435805767725, 14.01024538916436, 14.39757895580699, 14.75114708819012, 15.69896300880333, 16.21090077914068

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