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Properties

Label 2-483-23.22-c2-0-24
Degree $2$
Conductor $483$
Sign $0.950 + 0.311i$
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  − 0.327·2-s + 1.73·3-s − 3.89·4-s + 3.00i·5-s − 0.568·6-s − 2.64i·7-s + 2.58·8-s + 2.99·9-s − 0.986i·10-s − 9.21i·11-s − 6.74·12-s + 0.0704·13-s + 0.867i·14-s + 5.20i·15-s + 14.7·16-s + 4.13i·17-s + ⋯
L(s)  = 1  − 0.163·2-s + 0.577·3-s − 0.973·4-s + 0.601i·5-s − 0.0946·6-s − 0.377i·7-s + 0.323·8-s + 0.333·9-s − 0.0986i·10-s − 0.838i·11-s − 0.561·12-s + 0.00542·13-s + 0.0619i·14-s + 0.347i·15-s + 0.920·16-s + 0.243i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.950 + 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.950 + 0.311i$
Analytic conductor: \(13.1607\)
Root analytic conductor: \(3.62778\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1),\ 0.950 + 0.311i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.560376346\)
\(L(\frac12)\) \(\approx\) \(1.560376346\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
7 \( 1 + 2.64iT \)
23 \( 1 + (7.16 - 21.8i)T \)
good2 \( 1 + 0.327T + 4T^{2} \)
5 \( 1 - 3.00iT - 25T^{2} \)
11 \( 1 + 9.21iT - 121T^{2} \)
13 \( 1 - 0.0704T + 169T^{2} \)
17 \( 1 - 4.13iT - 289T^{2} \)
19 \( 1 + 18.2iT - 361T^{2} \)
29 \( 1 - 40.1T + 841T^{2} \)
31 \( 1 - 34.9T + 961T^{2} \)
37 \( 1 + 40.6iT - 1.36e3T^{2} \)
41 \( 1 - 74.0T + 1.68e3T^{2} \)
43 \( 1 - 27.9iT - 1.84e3T^{2} \)
47 \( 1 - 16.0T + 2.20e3T^{2} \)
53 \( 1 + 84.4iT - 2.80e3T^{2} \)
59 \( 1 - 49.2T + 3.48e3T^{2} \)
61 \( 1 + 8.08iT - 3.72e3T^{2} \)
67 \( 1 + 72.1iT - 4.48e3T^{2} \)
71 \( 1 + 14.2T + 5.04e3T^{2} \)
73 \( 1 + 120.T + 5.32e3T^{2} \)
79 \( 1 - 144. iT - 6.24e3T^{2} \)
83 \( 1 + 18.6iT - 6.88e3T^{2} \)
89 \( 1 + 127. iT - 7.92e3T^{2} \)
97 \( 1 - 90.3iT - 9.40e3T^{2} \)
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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

−10.56648422281418981089449374009, −9.772497736511358586383052151927, −8.897264435748788072665557192179, −8.171036502059997110138397006821, −7.24931064811338889513189145534, −6.11788952473598747491057123376, −4.82727217293897515855699492682, −3.80111553911602560324948257697, −2.78014426908887406542524813283, −0.847636046673767857328155984068, 1.10089466840694210398888231799, 2.71058933715819173277580293551, 4.23878467782855150621143890626, 4.80790385138309970779637355429, 6.08751805964686856136375294734, 7.45510302994768051052256258655, 8.415067528646258206890668266982, 8.859930019122476288813726159821, 9.858010295621266482843551798891, 10.38920316445634968272111217784

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