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Properties

Label 2-580-580.427-c0-0-0
Degree 22
Conductor 580580
Sign 0.997+0.0676i0.997 + 0.0676i
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  + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)4-s + (0.974 − 0.222i)5-s + (−0.222 + 0.974i)8-s + (0.222 − 0.974i)9-s + (−0.781 + 0.623i)10-s + (0.119 − 0.189i)13-s + (−0.222 − 0.974i)16-s − 0.867·17-s + (0.222 + 0.974i)18-s + (0.433 − 0.900i)20-s + (0.900 − 0.433i)25-s + (−0.0250 + 0.222i)26-s + (−0.974 + 0.222i)29-s + (0.623 + 0.781i)32-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (0.623 − 0.781i)4-s + (0.974 − 0.222i)5-s + (−0.222 + 0.974i)8-s + (0.222 − 0.974i)9-s + (−0.781 + 0.623i)10-s + (0.119 − 0.189i)13-s + (−0.222 − 0.974i)16-s − 0.867·17-s + (0.222 + 0.974i)18-s + (0.433 − 0.900i)20-s + (0.900 − 0.433i)25-s + (−0.0250 + 0.222i)26-s + (−0.974 + 0.222i)29-s + (0.623 + 0.781i)32-s + ⋯

Functional equation

Λ(s)=(580s/2 ΓC(s) L(s)=((0.997+0.0676i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0676i)\, \overline{\Lambda}(1-s) \end{aligned}
Λ(s)=(580s/2 ΓC(s) L(s)=((0.997+0.0676i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0676i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 580580    =    22⋅5⋅292^{2} \cdot 5 \cdot 29
Sign: 0.997+0.0676i0.997 + 0.0676i
Analytic conductor: 0.2894570.289457
Root analytic conductor: 0.5380120.538012
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ580(427,⋅)\chi_{580} (427, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 580, ( :0), 0.997+0.0676i)(2,\ 580,\ (\ :0),\ 0.997 + 0.0676i)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.71811606180.7181160618
L(12)L(\frac12) ≈\approx 0.71811606180.7181160618
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+(−0.974+0.222i)T 1 + (-0.974 + 0.222i)T
29 1+(0.974−0.222i)T 1 + (0.974 - 0.222i)T
good3 1+(−0.222+0.974i)T2 1 + (-0.222 + 0.974i)T^{2}
7 1+(0.974+0.222i)T2 1 + (0.974 + 0.222i)T^{2}
11 1+(−0.433−0.900i)T2 1 + (-0.433 - 0.900i)T^{2}
13 1+(−0.119+0.189i)T+(−0.433−0.900i)T2 1 + (-0.119 + 0.189i)T + (-0.433 - 0.900i)T^{2}
17 1+0.867T+T2 1 + 0.867T + T^{2}
19 1+(−0.974+0.222i)T2 1 + (-0.974 + 0.222i)T^{2}
23 1+(−0.781+0.623i)T2 1 + (-0.781 + 0.623i)T^{2}
31 1+(0.781+0.623i)T2 1 + (0.781 + 0.623i)T^{2}
37 1+(−1.21−0.277i)T+(0.900+0.433i)T2 1 + (-1.21 - 0.277i)T + (0.900 + 0.433i)T^{2}
41 1+(−1.40−1.40i)T+iT2 1 + (-1.40 - 1.40i)T + iT^{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+(0.623−1.78i)T+(−0.781−0.623i)T2 1 + (0.623 - 1.78i)T + (-0.781 - 0.623i)T^{2}
59 1+T2 1 + T^{2}
61 1+(−0.656−0.0739i)T+(0.974+0.222i)T2 1 + (-0.656 - 0.0739i)T + (0.974 + 0.222i)T^{2}
67 1+(0.433−0.900i)T2 1 + (0.433 - 0.900i)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+(−0.433+0.900i)T2 1 + (-0.433 + 0.900i)T^{2}
83 1+(0.974−0.222i)T2 1 + (0.974 - 0.222i)T^{2}
89 1+(1.00+0.351i)T+(0.781+0.623i)T2 1 + (1.00 + 0.351i)T + (0.781 + 0.623i)T^{2}
97 1+(1.22+0.974i)T+(0.222+0.974i)T2 1 + (1.22 + 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

−10.73562660683946360675082117585, −9.673054482159271619572933741290, −9.338953811063912605576300602229, −8.448368396388228754427793266674, −7.34653706057705363950325735902, −6.36442361073689490449061665002, −5.83578256167267737171192133503, −4.53434509519388340639219722607, −2.75062123749709962093078165071, −1.35795921922294321547072713129, 1.78752101784372838563881709381, 2.63904242292705613475051831101, 4.18041742797586052840275388863, 5.56262400445531744057112123902, 6.63310122311892209519594318311, 7.48210289346774768966317349248, 8.457382834439225665903971235690, 9.365695608946997222861982972484, 9.978771253867809022272812194734, 10.95074125898553509320008199060

Graph of the ZZ-function along the critical line