↑

Properties

Label 2-1287-1287.571-c0-0-4
Degree $2$
Conductor $1287$
Sign $-0.719 - 0.694i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.978 + 1.69i)2-s + (0.669 − 0.743i)3-s + (−1.41 + 2.44i)4-s + (1.91 + 0.406i)6-s + (0.809 + 1.40i)7-s − 3.57·8-s + (−0.104 − 0.994i)9-s + (−0.5 − 0.866i)11-s + (0.873 + 2.68i)12-s + (−0.5 + 0.866i)13-s + (−1.58 + 2.74i)14-s + (−2.08 − 3.60i)16-s + (1.58 − 1.14i)18-s + 1.33·19-s + (1.58 + 0.336i)21-s + (0.978 − 1.69i)22-s + ⋯
L(s)  = 1  + (0.978 + 1.69i)2-s + (0.669 − 0.743i)3-s + (−1.41 + 2.44i)4-s + (1.91 + 0.406i)6-s + (0.809 + 1.40i)7-s − 3.57·8-s + (−0.104 − 0.994i)9-s + (−0.5 − 0.866i)11-s + (0.873 + 2.68i)12-s + (−0.5 + 0.866i)13-s + (−1.58 + 2.74i)14-s + (−2.08 − 3.60i)16-s + (1.58 − 1.14i)18-s + 1.33·19-s + (1.58 + 0.336i)21-s + (0.978 − 1.69i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.719 - 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.719 - 0.694i$
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: \((2,\ 1287,\ (\ :0),\ -0.719 - 0.694i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.946742046\)
\(L(\frac12)\) \(\approx\) \(1.946742046\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.669 + 0.743i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (0.5 - 0.866i)T \)
good2 \( 1 + (-0.978 - 1.69i)T + (-0.5 + 0.866i)T^{2} \)
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 \( 1 - 1.33T + T^{2} \)
23 \( 1 + (-0.104 + 0.181i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.104 + 0.181i)T + (-0.5 - 0.866i)T^{2} \)
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 \( 1 + (0.913 + 1.58i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.522812868688311293759393297938, −8.827303311351950107965662127370, −8.278872309580605107897936300373, −7.67607008908906006852299763218, −6.87588740373776015854813874541, −5.95200585668628004263892459573, −5.43455842444764728026703786188, −4.47242202229215111207849284826, −3.26295510679496813472161724912, −2.38544164154186114454492017409, 1.34583239770082840794350654543, 2.50256565152095506107870179575, 3.45046154180415466605095186900, 4.15272498300767675797507722427, 4.98685647888127053319534614776, 5.40527481856236480030052741743, 7.27256371142684641852804937330, 8.038708307555564344750722181318, 9.329038044675233247630109147456, 9.918683882488226058231580181959

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