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Properties

Label 2-580-580.367-c0-0-0
Degree $2$
Conductor $580$
Sign $-0.968 - 0.249i$
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  + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.781 + 0.623i)5-s + (0.623 − 0.781i)8-s + (−0.623 + 0.781i)9-s + (−0.433 − 0.900i)10-s + (−0.211 + 1.87i)13-s + (0.623 + 0.781i)16-s − 1.94·17-s + (−0.623 − 0.781i)18-s + (0.974 − 0.222i)20-s + (0.222 − 0.974i)25-s + (−1.78 − 0.623i)26-s + (0.781 − 0.623i)29-s + (−0.900 + 0.433i)32-s + ⋯
L(s)  = 1  + (−0.222 + 0.974i)2-s + (−0.900 − 0.433i)4-s + (−0.781 + 0.623i)5-s + (0.623 − 0.781i)8-s + (−0.623 + 0.781i)9-s + (−0.433 − 0.900i)10-s + (−0.211 + 1.87i)13-s + (0.623 + 0.781i)16-s − 1.94·17-s + (−0.623 − 0.781i)18-s + (0.974 − 0.222i)20-s + (0.222 − 0.974i)25-s + (−1.78 − 0.623i)26-s + (0.781 − 0.623i)29-s + (−0.900 + 0.433i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.249i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(580\)    =    \(2^{2} \cdot 5 \cdot 29\)
Sign: $-0.968 - 0.249i$
Analytic conductor: \(0.289457\)
Root analytic conductor: \(0.538012\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{580} (367, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 580,\ (\ :0),\ -0.968 - 0.249i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4846324991\)
\(L(\frac12)\) \(\approx\) \(0.4846324991\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.222 - 0.974i)T \)
5 \( 1 + (0.781 - 0.623i)T \)
29 \( 1 + (-0.781 + 0.623i)T \)
good3 \( 1 + (0.623 - 0.781i)T^{2} \)
7 \( 1 + (-0.781 - 0.623i)T^{2} \)
11 \( 1 + (-0.974 - 0.222i)T^{2} \)
13 \( 1 + (0.211 - 1.87i)T + (-0.974 - 0.222i)T^{2} \)
17 \( 1 + 1.94T + T^{2} \)
19 \( 1 + (0.781 - 0.623i)T^{2} \)
23 \( 1 + (-0.433 - 0.900i)T^{2} \)
31 \( 1 + (0.433 - 0.900i)T^{2} \)
37 \( 1 + (-1.40 - 1.12i)T + (0.222 + 0.974i)T^{2} \)
41 \( 1 + (0.467 + 0.467i)T + iT^{2} \)
43 \( 1 + (-0.900 + 0.433i)T^{2} \)
47 \( 1 + (-0.222 + 0.974i)T^{2} \)
53 \( 1 + (-0.900 - 1.43i)T + (-0.433 + 0.900i)T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + (-0.351 + 1.00i)T + (-0.781 - 0.623i)T^{2} \)
67 \( 1 + (0.974 - 0.222i)T^{2} \)
71 \( 1 + (-0.222 + 0.974i)T^{2} \)
73 \( 1 + (0.0990 + 0.433i)T + (-0.900 + 0.433i)T^{2} \)
79 \( 1 + (-0.974 + 0.222i)T^{2} \)
83 \( 1 + (-0.781 + 0.623i)T^{2} \)
89 \( 1 + (-0.189 + 0.119i)T + (0.433 - 0.900i)T^{2} \)
97 \( 1 + (0.376 - 0.781i)T + (-0.623 - 0.781i)T^{2} \)
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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

−11.25002038462285750514454982661, −10.46836668788487525883502769882, −9.268151318834020937581226650674, −8.594963264572199101342003984695, −7.72229537938440201349109507118, −6.81638554299650766725216065758, −6.25426870938917399033602132525, −4.71764232881393486617445841441, −4.17026859678566591925847017438, −2.38262773568938247456531005224, 0.61338527950195630376983616365, 2.61365083104518836960731173802, 3.64511257994978781940458063528, 4.65033923956529695029546505796, 5.67026550816406635703881562448, 7.17276357695629184802268790366, 8.347014758150795561211928628421, 8.674915888081359923737009468550, 9.679861803907951591944330775340, 10.70552856138889021371757183155

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