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Properties

Label 2-1287-1287.571-c0-0-3
Degree $2$
Conductor $1287$
Sign $0.438 - 0.898i$
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  + (0.669 + 1.15i)2-s + (−0.978 + 0.207i)3-s + (−0.395 + 0.684i)4-s + (−0.895 − 0.994i)6-s + (−0.809 − 1.40i)7-s + 0.279·8-s + (0.913 − 0.406i)9-s + (0.5 + 0.866i)11-s + (0.244 − 0.752i)12-s + (0.5 − 0.866i)13-s + (1.08 − 1.87i)14-s + (0.582 + 1.00i)16-s + (1.08 + 0.786i)18-s + 1.95·19-s + (1.08 + 1.20i)21-s + (−0.669 + 1.15i)22-s + ⋯
L(s)  = 1  + (0.669 + 1.15i)2-s + (−0.978 + 0.207i)3-s + (−0.395 + 0.684i)4-s + (−0.895 − 0.994i)6-s + (−0.809 − 1.40i)7-s + 0.279·8-s + (0.913 − 0.406i)9-s + (0.5 + 0.866i)11-s + (0.244 − 0.752i)12-s + (0.5 − 0.866i)13-s + (1.08 − 1.87i)14-s + (0.582 + 1.00i)16-s + (1.08 + 0.786i)18-s + 1.95·19-s + (1.08 + 1.20i)21-s + (−0.669 + 1.15i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.438 - 0.898i$
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: \((2,\ 1287,\ (\ :0),\ 0.438 - 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.177477264\)
\(L(\frac12)\) \(\approx\) \(1.177477264\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.978 - 0.207i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (-0.669 - 1.15i)T + (-0.5 + 0.866i)T^{2} \)
5 \( 1 + (0.5 + 0.866i)T^{2} \)
7 \( 1 + (0.809 + 1.40i)T + (-0.5 + 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - 1.95T + T^{2} \)
23 \( 1 + (0.913 - 1.58i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.913 + 1.58i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 - 0.618T + T^{2} \)
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 \( 1 + 0.618T + T^{2} \)
79 \( 1 + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.104 + 0.181i)T + (-0.5 + 0.866i)T^{2} \)
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

−10.07458141569923431940930030415, −9.427837952711917449843891119929, −7.67359676354091322403551075742, −7.41721046689799662126497053239, −6.62948912757454750011379014498, −5.83439965208009665607168362317, −5.21389808921521144623745163808, −4.07358340826310069895164182385, −3.66911727605133943935336521022, −1.19864659852857288470330643480, 1.33185063085812467798762461849, 2.55482670968959648255201064366, 3.49497865283038354677690618182, 4.47947281558384391009246130734, 5.56050820314487710488745759182, 6.04514811183859271015095924630, 6.97497955481700610962921103452, 8.177164863238303401839655672886, 9.328173124767174210736024174203, 9.807673241033242775668296905536

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