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Properties

Label 2-1287-1287.571-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(0.8210286888\)
\(L(\frac12)\) \(\approx\) \(0.8210286888\)
\(L(1)\) 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

−9.794462920488303623500360124282, −8.752514260786674733772215146248, −8.375956581134059663159555785259, −7.31860851631546025744466348630, −6.17263016718159237715365301768, −5.64896942522348988063839843898, −4.77412569709562646225241726885, −2.90604918736152011723007270213, −2.16051852425647390494265817457, −1.26394259269510389978642976286, 1.13847392226770135971378215185, 3.47480962363291170688952117094, 3.95522946833482582291930552116, 5.12200633924329992388128251392, 5.94357438983016738342417216607, 6.99767094037660796518173280472, 7.41241232742457990445604674169, 8.465570964051848098140038582638, 9.091281674116506414438019337703, 9.938889609463993239076139993217

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