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Properties

Label 2-580-580.367-c0-0-0
Degree 22
Conductor 580580
Sign −0.968−0.249i-0.968 - 0.249i
Analytic cond. 0.2894570.289457
Root an. cond. 0.5380120.538012
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

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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

Λ(s)=(580s/2 ΓC(s) L(s)=((−0.968−0.249i) Λ‾(1−s)\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}
Λ(s)=(580s/2 ΓC(s) L(s)=((−0.968−0.249i) Λ‾(1−s)\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: 22
Conductor: 580580    =    22⋅5⋅292^{2} \cdot 5 \cdot 29
Sign: −0.968−0.249i-0.968 - 0.249i
Analytic conductor: 0.2894570.289457
Root analytic conductor: 0.5380120.538012
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ580(367,⋅)\chi_{580} (367, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 580, ( :0), −0.968−0.249i)(2,\ 580,\ (\ :0),\ -0.968 - 0.249i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.48463249910.4846324991
L(12)L(\frac12) ≈\approx 0.48463249910.4846324991
L(1)L(1) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1+(0.222−0.974i)T 1 + (0.222 - 0.974i)T
5 1+(0.781−0.623i)T 1 + (0.781 - 0.623i)T
29 1+(−0.781+0.623i)T 1 + (-0.781 + 0.623i)T
good3 1+(0.623−0.781i)T2 1 + (0.623 - 0.781i)T^{2}
7 1+(−0.781−0.623i)T2 1 + (-0.781 - 0.623i)T^{2}
11 1+(−0.974−0.222i)T2 1 + (-0.974 - 0.222i)T^{2}
13 1+(0.211−1.87i)T+(−0.974−0.222i)T2 1 + (0.211 - 1.87i)T + (-0.974 - 0.222i)T^{2}
17 1+1.94T+T2 1 + 1.94T + T^{2}
19 1+(0.781−0.623i)T2 1 + (0.781 - 0.623i)T^{2}
23 1+(−0.433−0.900i)T2 1 + (-0.433 - 0.900i)T^{2}
31 1+(0.433−0.900i)T2 1 + (0.433 - 0.900i)T^{2}
37 1+(−1.40−1.12i)T+(0.222+0.974i)T2 1 + (-1.40 - 1.12i)T + (0.222 + 0.974i)T^{2}
41 1+(0.467+0.467i)T+iT2 1 + (0.467 + 0.467i)T + iT^{2}
43 1+(−0.900+0.433i)T2 1 + (-0.900 + 0.433i)T^{2}
47 1+(−0.222+0.974i)T2 1 + (-0.222 + 0.974i)T^{2}
53 1+(−0.900−1.43i)T+(−0.433+0.900i)T2 1 + (-0.900 - 1.43i)T + (-0.433 + 0.900i)T^{2}
59 1+T2 1 + T^{2}
61 1+(−0.351+1.00i)T+(−0.781−0.623i)T2 1 + (-0.351 + 1.00i)T + (-0.781 - 0.623i)T^{2}
67 1+(0.974−0.222i)T2 1 + (0.974 - 0.222i)T^{2}
71 1+(−0.222+0.974i)T2 1 + (-0.222 + 0.974i)T^{2}
73 1+(0.0990+0.433i)T+(−0.900+0.433i)T2 1 + (0.0990 + 0.433i)T + (-0.900 + 0.433i)T^{2}
79 1+(−0.974+0.222i)T2 1 + (-0.974 + 0.222i)T^{2}
83 1+(−0.781+0.623i)T2 1 + (-0.781 + 0.623i)T^{2}
89 1+(−0.189+0.119i)T+(0.433−0.900i)T2 1 + (-0.189 + 0.119i)T + (0.433 - 0.900i)T^{2}
97 1+(0.376−0.781i)T+(−0.623−0.781i)T2 1 + (0.376 - 0.781i)T + (-0.623 - 0.781i)T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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 ZZ-function along the critical line