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Properties

Label 2-483-21.20-c1-0-47
Degree $2$
Conductor $483$
Sign $0.662 + 0.749i$
Analytic cond. $3.85677$
Root an. cond. $1.96386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

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

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

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.662 + 0.749i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.662 + 0.749i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00431 - 0.903121i\)
\(L(\frac12)\) \(\approx\) \(2.00431 - 0.903121i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.33 + 1.09i)T \)
7 \( 1 + (2.64 - 0.0997i)T \)
23 \( 1 - iT \)
good2 \( 1 + 0.123iT - 2T^{2} \)
5 \( 1 - 0.867T + 5T^{2} \)
11 \( 1 - 3.54iT - 11T^{2} \)
13 \( 1 + 5.96iT - 13T^{2} \)
17 \( 1 - 4.76T + 17T^{2} \)
19 \( 1 + 0.890iT - 19T^{2} \)
29 \( 1 - 7.97iT - 29T^{2} \)
31 \( 1 - 5.07iT - 31T^{2} \)
37 \( 1 + 7.17T + 37T^{2} \)
41 \( 1 - 9.57T + 41T^{2} \)
43 \( 1 - 1.82T + 43T^{2} \)
47 \( 1 + 12.0T + 47T^{2} \)
53 \( 1 - 8.39iT - 53T^{2} \)
59 \( 1 + 7.95T + 59T^{2} \)
61 \( 1 - 5.38iT - 61T^{2} \)
67 \( 1 + 3.03T + 67T^{2} \)
71 \( 1 + 3.17iT - 71T^{2} \)
73 \( 1 + 9.86iT - 73T^{2} \)
79 \( 1 - 3.97T + 79T^{2} \)
83 \( 1 + 1.40T + 83T^{2} \)
89 \( 1 + 9.80T + 89T^{2} \)
97 \( 1 - 12.7iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

Graph of the $Z$-function along the critical line