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Properties

Label 2-580-580.419-c0-0-2
Degree 22
Conductor 580580
Sign −0.00819+0.999i-0.00819 + 0.999i
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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)4-s + (0.222 − 0.974i)5-s + (−0.222 − 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.623 + 0.781i)10-s + (−1.52 − 0.347i)13-s + (−0.222 + 0.974i)16-s + 1.80·17-s + (−0.222 + 0.974i)18-s + (0.900 − 0.433i)20-s + (−0.900 − 0.433i)25-s + (1.22 + 0.974i)26-s + (0.222 − 0.974i)29-s + (0.623 − 0.781i)32-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (0.623 + 0.781i)4-s + (0.222 − 0.974i)5-s + (−0.222 − 0.974i)8-s + (−0.222 − 0.974i)9-s + (−0.623 + 0.781i)10-s + (−1.52 − 0.347i)13-s + (−0.222 + 0.974i)16-s + 1.80·17-s + (−0.222 + 0.974i)18-s + (0.900 − 0.433i)20-s + (−0.900 − 0.433i)25-s + (1.22 + 0.974i)26-s + (0.222 − 0.974i)29-s + (0.623 − 0.781i)32-s + ⋯

Functional equation

Λ(s)=(580s/2 ΓC(s) L(s)=((−0.00819+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00819 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(580s/2 ΓC(s) L(s)=((−0.00819+0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00819 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 580580    =    22⋅5⋅292^{2} \cdot 5 \cdot 29
Sign: −0.00819+0.999i-0.00819 + 0.999i
Analytic conductor: 0.2894570.289457
Root analytic conductor: 0.5380120.538012
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ580(419,⋅)\chi_{580} (419, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 580, ( :0), −0.00819+0.999i)(2,\ 580,\ (\ :0),\ -0.00819 + 0.999i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.59167110860.5916711086
L(12)L(\frac12) ≈\approx 0.59167110860.5916711086
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+(0.900+0.433i)T 1 + (0.900 + 0.433i)T
5 1+(−0.222+0.974i)T 1 + (-0.222 + 0.974i)T
29 1+(−0.222+0.974i)T 1 + (-0.222 + 0.974i)T
good3 1+(0.222+0.974i)T2 1 + (0.222 + 0.974i)T^{2}
7 1+(−0.222−0.974i)T2 1 + (-0.222 - 0.974i)T^{2}
11 1+(−0.900−0.433i)T2 1 + (-0.900 - 0.433i)T^{2}
13 1+(1.52+0.347i)T+(0.900+0.433i)T2 1 + (1.52 + 0.347i)T + (0.900 + 0.433i)T^{2}
17 1−1.80T+T2 1 - 1.80T + T^{2}
19 1+(−0.222+0.974i)T2 1 + (-0.222 + 0.974i)T^{2}
23 1+(0.623−0.781i)T2 1 + (0.623 - 0.781i)T^{2}
31 1+(0.623+0.781i)T2 1 + (0.623 + 0.781i)T^{2}
37 1+(−0.277−1.21i)T+(−0.900+0.433i)T2 1 + (-0.277 - 1.21i)T + (-0.900 + 0.433i)T^{2}
41 1+1.56iT−T2 1 + 1.56iT - T^{2}
43 1+(−0.623+0.781i)T2 1 + (-0.623 + 0.781i)T^{2}
47 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
53 1+(0.376−0.781i)T+(−0.623−0.781i)T2 1 + (0.376 - 0.781i)T + (-0.623 - 0.781i)T^{2}
59 1−T2 1 - T^{2}
61 1+(−0.678−0.541i)T+(0.222+0.974i)T2 1 + (-0.678 - 0.541i)T + (0.222 + 0.974i)T^{2}
67 1+(−0.900+0.433i)T2 1 + (-0.900 + 0.433i)T^{2}
71 1+(0.900+0.433i)T2 1 + (0.900 + 0.433i)T^{2}
73 1+(−1.62+0.781i)T+(0.623−0.781i)T2 1 + (-1.62 + 0.781i)T + (0.623 - 0.781i)T^{2}
79 1+(−0.900+0.433i)T2 1 + (-0.900 + 0.433i)T^{2}
83 1+(−0.222+0.974i)T2 1 + (-0.222 + 0.974i)T^{2}
89 1+(0.846−1.75i)T+(−0.623−0.781i)T2 1 + (0.846 - 1.75i)T + (-0.623 - 0.781i)T^{2}
97 1+(−0.777−0.974i)T+(−0.222+0.974i)T2 1 + (-0.777 - 0.974i)T + (-0.222 + 0.974i)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

−10.38560838895886154265253898632, −9.710193262600687845961269645002, −9.202172644692105682967168649472, −8.127608055746079847358013667722, −7.50202937962206203170318903768, −6.24545276788656037600926472239, −5.17128142779901383393917760622, −3.80899876621016052401476391686, −2.58036098374940259624040257295, −0.974021854137760207987405661264, 1.99511065622949100237121090086, 3.04840576461301472614308537612, 4.99688619289435750714136435077, 5.79517032363286600231751767790, 6.99456366694300532253474970825, 7.52632301776874556676190379813, 8.337621867457639573405292453998, 9.726693322343953261540183537096, 9.988917598154381663673301888891, 10.93595239171206363730687515435

Graph of the ZZ-function along the critical line