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Properties

Label 2-1287-1287.571-c0-0-9
Degree $2$
Conductor $1287$
Sign $0.173 + 0.984i$
Analytic cond. $0.642296$
Root an. cond. $0.801434$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)6-s + (−1 − 1.73i)7-s + 8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)13-s + (0.999 − 1.73i)14-s + (0.5 + 0.866i)16-s − 0.999·18-s − 19-s + (−0.999 + 1.73i)21-s + (0.499 − 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)24-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 − 0.866i)3-s + (0.499 − 0.866i)6-s + (−1 − 1.73i)7-s + 8-s + (−0.499 + 0.866i)9-s + (−0.5 − 0.866i)11-s + (−0.5 + 0.866i)13-s + (0.999 − 1.73i)14-s + (0.5 + 0.866i)16-s − 0.999·18-s − 19-s + (−0.999 + 1.73i)21-s + (0.499 − 0.866i)22-s + (0.5 − 0.866i)23-s + (−0.499 − 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.173 + 0.984i$
Analytic conductor: \(0.642296\)
Root analytic conductor: \(0.801434\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (571, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1287,\ (\ :0),\ 0.173 + 0.984i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9625544604\)
\(L(\frac12)\) \(\approx\) \(0.9625544604\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T + T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - 2T + T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 2T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
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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

−9.882048810598072737327855799302, −8.506018451468121566279352820388, −7.64579325184881463344749175441, −6.92392590352739902872418249663, −6.55690018311463055675949562023, −5.80244561906078112627797731325, −4.69008591517643597235730218490, −3.90109892147256557446574960322, −2.33820889246627144752961003966, −0.71617514634677239888017778061, 2.24137773676379497841388803923, 2.96490615798037802139686232564, 3.82097787903851002562791649832, 5.01114671762773238081295657025, 5.49290995524950370226918996633, 6.49155146565755211895100090737, 7.61859221204902879203889809373, 8.722263315612160654985535343368, 9.568407969219074244996474532165, 10.06163680933256172038109164871

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