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Properties

Label 2-579-579.470-c0-0-1
Degree $2$
Conductor $579$
Sign $0.319 - 0.947i$
Analytic cond. $0.288958$
Root an. cond. $0.537548$
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  + (−0.866 + 0.5i)2-s − 3-s + (0.866 − 0.5i)5-s + (0.866 − 0.5i)6-s + (−0.5 + 0.866i)7-s − i·8-s + 9-s + (−0.499 + 0.866i)10-s − 0.999i·14-s + (−0.866 + 0.5i)15-s + (0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.866 + 0.5i)18-s + (0.5 + 0.866i)19-s + (0.5 − 0.866i)21-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)2-s − 3-s + (0.866 − 0.5i)5-s + (0.866 − 0.5i)6-s + (−0.5 + 0.866i)7-s − i·8-s + 9-s + (−0.499 + 0.866i)10-s − 0.999i·14-s + (−0.866 + 0.5i)15-s + (0.5 + 0.866i)16-s + (0.866 − 0.5i)17-s + (−0.866 + 0.5i)18-s + (0.5 + 0.866i)19-s + (0.5 − 0.866i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 579 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(579\)    =    \(3 \cdot 193\)
Sign: $0.319 - 0.947i$
Analytic conductor: \(0.288958\)
Root analytic conductor: \(0.537548\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{579} (470, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 579,\ (\ :0),\ 0.319 - 0.947i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4563822203\)
\(L(\frac12)\) \(\approx\) \(0.4563822203\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
193 \( 1 + T \)
good2 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - 2iT - T^{2} \)
31 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + T^{2} \)
71 \( 1 + 2iT - T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
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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

−10.87278502087030210025405018618, −9.842803868289167462882566101994, −9.503743052541885024849947232777, −8.603511989446248120607949056367, −7.51156353969804382319455899466, −6.58952151838053590003957246389, −5.73598951446830941738461399646, −5.02220858443781203877800749982, −3.39059255345849973519221049367, −1.39868853194480044880100706238, 0.959017651599993919959886094328, 2.44821514753974454010548095193, 4.13984546594890321006636282303, 5.43958187678317832063951433259, 6.17928512076420945251274456006, 7.12522532715940480102686783858, 8.166815993278388741795577976385, 9.651863636674516439073334317404, 9.908172116709605009313193241194, 10.51359087647661724259316799327

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