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Properties

Label 2-483-21.20-c1-0-28
Degree 22
Conductor 483483
Sign 0.662−0.749i0.662 - 0.749i
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  + 0.123i·2-s + (1.33 + 1.09i)3-s + 1.98·4-s + 0.867·5-s + (−0.135 + 0.165i)6-s + (−2.64 − 0.0997i)7-s + 0.491i·8-s + (0.590 + 2.94i)9-s + 0.107i·10-s − 3.54i·11-s + (2.65 + 2.17i)12-s + 5.96i·13-s + (0.0123 − 0.326i)14-s + (1.16 + 0.951i)15-s + 3.90·16-s + 4.76·17-s + ⋯
L(s)  = 1  + 0.0872i·2-s + (0.773 + 0.633i)3-s + 0.992·4-s + 0.387·5-s + (−0.0553 + 0.0675i)6-s + (−0.999 − 0.0377i)7-s + 0.173i·8-s + (0.196 + 0.980i)9-s + 0.0338i·10-s − 1.07i·11-s + (0.767 + 0.628i)12-s + 1.65i·13-s + (0.00329 − 0.0872i)14-s + (0.299 + 0.245i)15-s + 0.977·16-s + 1.15·17-s + ⋯

Functional equation

Λ(s)=(483s/2 ΓC(s) L(s)=((0.662−0.749i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(483s/2 ΓC(s+1/2) L(s)=((0.662−0.749i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 483483    =    3⋅7⋅233 \cdot 7 \cdot 23
Sign: 0.662−0.749i0.662 - 0.749i
Analytic conductor: 3.856773.85677
Root analytic conductor: 1.963861.96386
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ483(461,⋅)\chi_{483} (461, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 483, ( :1/2), 0.662−0.749i)(2,\ 483,\ (\ :1/2),\ 0.662 - 0.749i)

Particular Values

L(1)L(1) ≈\approx 2.00431+0.903121i2.00431 + 0.903121i
L(12)L(\frac12) ≈\approx 2.00431+0.903121i2.00431 + 0.903121i
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+(−1.33−1.09i)T 1 + (-1.33 - 1.09i)T
7 1+(2.64+0.0997i)T 1 + (2.64 + 0.0997i)T
23 1+iT 1 + iT
good2 1−0.123iT−2T2 1 - 0.123iT - 2T^{2}
5 1−0.867T+5T2 1 - 0.867T + 5T^{2}
11 1+3.54iT−11T2 1 + 3.54iT - 11T^{2}
13 1−5.96iT−13T2 1 - 5.96iT - 13T^{2}
17 1−4.76T+17T2 1 - 4.76T + 17T^{2}
19 1−0.890iT−19T2 1 - 0.890iT - 19T^{2}
29 1+7.97iT−29T2 1 + 7.97iT - 29T^{2}
31 1+5.07iT−31T2 1 + 5.07iT - 31T^{2}
37 1+7.17T+37T2 1 + 7.17T + 37T^{2}
41 1−9.57T+41T2 1 - 9.57T + 41T^{2}
43 1−1.82T+43T2 1 - 1.82T + 43T^{2}
47 1+12.0T+47T2 1 + 12.0T + 47T^{2}
53 1+8.39iT−53T2 1 + 8.39iT - 53T^{2}
59 1+7.95T+59T2 1 + 7.95T + 59T^{2}
61 1+5.38iT−61T2 1 + 5.38iT - 61T^{2}
67 1+3.03T+67T2 1 + 3.03T + 67T^{2}
71 1−3.17iT−71T2 1 - 3.17iT - 71T^{2}
73 1−9.86iT−73T2 1 - 9.86iT - 73T^{2}
79 1−3.97T+79T2 1 - 3.97T + 79T^{2}
83 1+1.40T+83T2 1 + 1.40T + 83T^{2}
89 1+9.80T+89T2 1 + 9.80T + 89T^{2}
97 1+12.7iT−97T2 1 + 12.7iT - 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

−11.08308825343362478535655495646, −9.924105751565373935333801136161, −9.591268228803492918213581993811, −8.429854044363331841364123649454, −7.51047003343886276897543866442, −6.40608970490427444782622142675, −5.69839659711224474409497011712, −4.02624939365630349473202236292, −3.11806389448548128571118386462, −1.99662701512959831924472491482, 1.45970608384783667698042817853, 2.80380859832759550549705653560, 3.42926084528796805027780531207, 5.47237001086774653495016474758, 6.39233687221323089789589237267, 7.30754119328450489361098869332, 7.85549971918272669572453728350, 9.184704371224389326167726204748, 10.02291281754907214010740608634, 10.60888051339957613769486382115

Graph of the ZZ-function along the critical line