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Properties

Label 2-483-7.6-c2-0-13
Degree $2$
Conductor $483$
Sign $0.785 - 0.618i$
Analytic cond. $13.1607$
Root an. cond. $3.62778$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.13·2-s − 1.73i·3-s + 5.82·4-s − 7.50i·5-s + 5.42i·6-s + (4.32 + 5.50i)7-s − 5.70·8-s − 2.99·9-s + 23.5i·10-s + 11.2·11-s − 10.0i·12-s + 4.27i·13-s + (−13.5 − 17.2i)14-s − 12.9·15-s − 5.39·16-s + 27.9i·17-s + ⋯
L(s)  = 1  − 1.56·2-s − 0.577i·3-s + 1.45·4-s − 1.50i·5-s + 0.904i·6-s + (0.618 + 0.785i)7-s − 0.713·8-s − 0.333·9-s + 2.35i·10-s + 1.02·11-s − 0.840i·12-s + 0.328i·13-s + (−0.968 − 1.23i)14-s − 0.866·15-s − 0.337·16-s + 1.64i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.785 - 0.618i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.785 - 0.618i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.785 - 0.618i$
Analytic conductor: \(13.1607\)
Root analytic conductor: \(3.62778\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1),\ 0.785 - 0.618i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.6685569881\)
\(L(\frac12)\) \(\approx\) \(0.6685569881\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 1.73iT \)
7 \( 1 + (-4.32 - 5.50i)T \)
23 \( 1 - 4.79T \)
good2 \( 1 + 3.13T + 4T^{2} \)
5 \( 1 + 7.50iT - 25T^{2} \)
11 \( 1 - 11.2T + 121T^{2} \)
13 \( 1 - 4.27iT - 169T^{2} \)
17 \( 1 - 27.9iT - 289T^{2} \)
19 \( 1 - 34.7iT - 361T^{2} \)
29 \( 1 + 43.1T + 841T^{2} \)
31 \( 1 - 26.4iT - 961T^{2} \)
37 \( 1 + 10.1T + 1.36e3T^{2} \)
41 \( 1 - 19.2iT - 1.68e3T^{2} \)
43 \( 1 + 33.6T + 1.84e3T^{2} \)
47 \( 1 - 12.3iT - 2.20e3T^{2} \)
53 \( 1 + 27.7T + 2.80e3T^{2} \)
59 \( 1 - 73.4iT - 3.48e3T^{2} \)
61 \( 1 - 65.6iT - 3.72e3T^{2} \)
67 \( 1 - 3.99T + 4.48e3T^{2} \)
71 \( 1 - 104.T + 5.04e3T^{2} \)
73 \( 1 - 29.8iT - 5.32e3T^{2} \)
79 \( 1 - 72.3T + 6.24e3T^{2} \)
83 \( 1 + 70.4iT - 6.88e3T^{2} \)
89 \( 1 - 35.0iT - 7.92e3T^{2} \)
97 \( 1 - 18.2iT - 9.40e3T^{2} \)
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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

−10.73233044586117185891078666208, −9.603309187560576970286500744101, −8.871656514061344680044056516082, −8.380744109219919385707626132479, −7.76222237959072555969156222866, −6.41310755115249604977538643086, −5.47234994626345166327642813333, −4.05579978957133313925962982959, −1.66249067344586789195542012592, −1.45182842114690138727155661336, 0.48812563552175704360275550889, 2.26998651302612048287642788090, 3.51677862098776513149435431201, 4.91525407990837789020084863777, 6.69299328477953224137125253590, 7.10933434340342070240069145538, 7.960261020511909354080108010267, 9.250023438424274692415727530656, 9.622191913276438217274193555589, 10.66332556954165376276335467967

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