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Properties

Label 2-483-69.68-c1-0-23
Degree 22
Conductor 483483
Sign 0.897−0.441i0.897 - 0.441i
Analytic cond. 3.856773.85677
Root an. cond. 1.963861.96386
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.369i·2-s + (0.867 + 1.49i)3-s + 1.86·4-s − 1.03·5-s + (0.554 − 0.320i)6-s − i·7-s − 1.42i·8-s + (−1.49 + 2.60i)9-s + 0.381i·10-s + 0.430·11-s + (1.61 + 2.79i)12-s + 4.27·13-s − 0.369·14-s + (−0.895 − 1.54i)15-s + 3.19·16-s + 7.25·17-s + ⋯
L(s)  = 1  − 0.261i·2-s + (0.500 + 0.865i)3-s + 0.931·4-s − 0.461·5-s + (0.226 − 0.130i)6-s − 0.377i·7-s − 0.504i·8-s + (−0.498 + 0.866i)9-s + 0.120i·10-s + 0.129·11-s + (0.466 + 0.806i)12-s + 1.18·13-s − 0.0987·14-s + (−0.231 − 0.399i)15-s + 0.799·16-s + 1.75·17-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=((0.897−0.441i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1/2) L(s)=((0.897−0.441i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: 0.897−0.441i0.897 - 0.441i
Analytic conductor: 3.856773.85677
Root analytic conductor: 1.963861.96386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ483(344,⋅)\chi_{483} (344, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 483, ( :1/2), 0.897−0.441i)(2,\ 483,\ (\ :1/2),\ 0.897 - 0.441i)

Particular Values

L(1)L(1) ≈\approx 1.92780+0.448621i1.92780 + 0.448621i
L(12)L(\frac12) ≈\approx 1.92780+0.448621i1.92780 + 0.448621i
L(32)L(\frac{3}{2}) 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)
bad3 1+(−0.867−1.49i)T 1 + (-0.867 - 1.49i)T
7 1+iT 1 + iT
23 1+(4.78+0.321i)T 1 + (4.78 + 0.321i)T
good2 1+0.369iT−2T2 1 + 0.369iT - 2T^{2}
5 1+1.03T+5T2 1 + 1.03T + 5T^{2}
11 1−0.430T+11T2 1 - 0.430T + 11T^{2}
13 1−4.27T+13T2 1 - 4.27T + 13T^{2}
17 1−7.25T+17T2 1 - 7.25T + 17T^{2}
19 1−3.53iT−19T2 1 - 3.53iT - 19T^{2}
29 1−4.52iT−29T2 1 - 4.52iT - 29T^{2}
31 1+3.23T+31T2 1 + 3.23T + 31T^{2}
37 1−8.22iT−37T2 1 - 8.22iT - 37T^{2}
41 1+4.85iT−41T2 1 + 4.85iT - 41T^{2}
43 1+9.90iT−43T2 1 + 9.90iT - 43T^{2}
47 1+0.312iT−47T2 1 + 0.312iT - 47T^{2}
53 1+10.6T+53T2 1 + 10.6T + 53T^{2}
59 1+8.46iT−59T2 1 + 8.46iT - 59T^{2}
61 1+11.8iT−61T2 1 + 11.8iT - 61T^{2}
67 1+5.50iT−67T2 1 + 5.50iT - 67T^{2}
71 1+12.5iT−71T2 1 + 12.5iT - 71T^{2}
73 1+4.83T+73T2 1 + 4.83T + 73T^{2}
79 1−17.2iT−79T2 1 - 17.2iT - 79T^{2}
83 1−10.7T+83T2 1 - 10.7T + 83T^{2}
89 1−4.36T+89T2 1 - 4.36T + 89T^{2}
97 1−4.63iT−97T2 1 - 4.63iT - 97T^{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

−10.87322720856764381812012155290, −10.30550044698311679803843242870, −9.501343602944300677736509871803, −8.149212034506522385082309214831, −7.74139156340489728631596061040, −6.36814188984852902671178252125, −5.35663301284854156199286993950, −3.69744516547076024483326943872, −3.46099870738988631898592723584, −1.70753060696453066544896145013, 1.41579307779124018092319281243, 2.75187293605108719694838604498, 3.78236785221669010353018103717, 5.76622080477071335345063218797, 6.25011316298649649213292833363, 7.52903191162504721613827519451, 7.86604791310157054876219015164, 8.858170132576418900798716113669, 9.990371343057969038729689275379, 11.28015702902507735131772876337

Graph of the ZZ-function along the critical line