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Properties

Label 2-580-580.299-c0-0-0
Degree $2$
Conductor $580$
Sign $-0.776 + 0.629i$
Analytic cond. $0.289457$
Root an. cond. $0.538012$
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  + (0.433 + 0.900i)2-s + (−1.40 + 1.12i)3-s + (−0.623 + 0.781i)4-s + (−0.900 + 0.433i)5-s + (−1.62 − 0.781i)6-s + (0.974 + 1.22i)7-s + (−0.974 − 0.222i)8-s + (0.500 − 2.19i)9-s + (−0.781 − 0.623i)10-s − 1.80i·12-s + (−0.678 + 1.40i)14-s + (0.781 − 1.62i)15-s + (−0.222 − 0.974i)16-s + (2.19 − 0.499i)18-s + (0.222 − 0.974i)20-s + (−2.74 − 0.626i)21-s + ⋯
L(s)  = 1  + (0.433 + 0.900i)2-s + (−1.40 + 1.12i)3-s + (−0.623 + 0.781i)4-s + (−0.900 + 0.433i)5-s + (−1.62 − 0.781i)6-s + (0.974 + 1.22i)7-s + (−0.974 − 0.222i)8-s + (0.500 − 2.19i)9-s + (−0.781 − 0.623i)10-s − 1.80i·12-s + (−0.678 + 1.40i)14-s + (0.781 − 1.62i)15-s + (−0.222 − 0.974i)16-s + (2.19 − 0.499i)18-s + (0.222 − 0.974i)20-s + (−2.74 − 0.626i)21-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.776 + 0.629i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(580\)    =    \(2^{2} \cdot 5 \cdot 29\)
Sign: $-0.776 + 0.629i$
Analytic conductor: \(0.289457\)
Root analytic conductor: \(0.538012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{580} (299, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 580,\ (\ :0),\ -0.776 + 0.629i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5338538972\)
\(L(\frac12)\) \(\approx\) \(0.5338538972\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.433 - 0.900i)T \)
5 \( 1 + (0.900 - 0.433i)T \)
29 \( 1 + (-0.623 - 0.781i)T \)
good3 \( 1 + (1.40 - 1.12i)T + (0.222 - 0.974i)T^{2} \)
7 \( 1 + (-0.974 - 1.22i)T + (-0.222 + 0.974i)T^{2} \)
11 \( 1 + (-0.900 + 0.433i)T^{2} \)
13 \( 1 + (0.900 - 0.433i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-0.222 - 0.974i)T^{2} \)
23 \( 1 + (0.781 + 0.376i)T + (0.623 + 0.781i)T^{2} \)
31 \( 1 + (0.623 - 0.781i)T^{2} \)
37 \( 1 + (-0.900 - 0.433i)T^{2} \)
41 \( 1 - 1.94iT - T^{2} \)
43 \( 1 + (0.193 - 0.400i)T + (-0.623 - 0.781i)T^{2} \)
47 \( 1 + (1.21 - 0.277i)T + (0.900 - 0.433i)T^{2} \)
53 \( 1 + (-0.623 + 0.781i)T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + (0.222 - 0.974i)T^{2} \)
67 \( 1 + (-0.900 - 0.433i)T^{2} \)
71 \( 1 + (0.900 - 0.433i)T^{2} \)
73 \( 1 + (0.623 + 0.781i)T^{2} \)
79 \( 1 + (-0.900 - 0.433i)T^{2} \)
83 \( 1 + (-1.21 + 1.52i)T + (-0.222 - 0.974i)T^{2} \)
89 \( 1 + (0.678 + 1.40i)T + (-0.623 + 0.781i)T^{2} \)
97 \( 1 + (-0.222 - 0.974i)T^{2} \)
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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

−11.68733627104181718128572090911, −10.80032210254264957931007956670, −9.728416299830739670371558354656, −8.678734606716929004442161415362, −7.907079324467634884328553916231, −6.60706318046680288153931745394, −5.91288713015033663805543422607, −4.91019011840062813779091779502, −4.48365809669072347024963088760, −3.21768456176988048915957176451, 0.68914589525004690176147819985, 1.80439613628457985677589988269, 3.91017569259715494344128131025, 4.74227012005394040943169947429, 5.54193691621917739086766506809, 6.75030281757588365054569043936, 7.62722806270351580286674733121, 8.396711999859045673390622532596, 10.04388673675057293031320978471, 10.90306074982742365940136348349

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