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Properties

Label 2-483-21.20-c1-0-32
Degree $2$
Conductor $483$
Sign $0.698 + 0.715i$
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  − 0.549i·2-s + (−1.68 − 0.420i)3-s + 1.69·4-s + 3.60·5-s + (−0.230 + 0.922i)6-s + (1.38 − 2.25i)7-s − 2.03i·8-s + (2.64 + 1.41i)9-s − 1.98i·10-s + 5.41i·11-s + (−2.85 − 0.714i)12-s − 1.72i·13-s + (−1.23 − 0.761i)14-s + (−6.06 − 1.51i)15-s + 2.28·16-s − 4.50·17-s + ⋯
L(s)  = 1  − 0.388i·2-s + (−0.970 − 0.242i)3-s + 0.849·4-s + 1.61·5-s + (−0.0942 + 0.376i)6-s + (0.524 − 0.851i)7-s − 0.717i·8-s + (0.882 + 0.471i)9-s − 0.626i·10-s + 1.63i·11-s + (−0.823 − 0.206i)12-s − 0.479i·13-s + (−0.330 − 0.203i)14-s + (−1.56 − 0.391i)15-s + 0.570·16-s − 1.09·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.698 + 0.715i$
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: \((2,\ 483,\ (\ :1/2),\ 0.698 + 0.715i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61601 - 0.680581i\)
\(L(\frac12)\) \(\approx\) \(1.61601 - 0.680581i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.68 + 0.420i)T \)
7 \( 1 + (-1.38 + 2.25i)T \)
23 \( 1 - iT \)
good2 \( 1 + 0.549iT - 2T^{2} \)
5 \( 1 - 3.60T + 5T^{2} \)
11 \( 1 - 5.41iT - 11T^{2} \)
13 \( 1 + 1.72iT - 13T^{2} \)
17 \( 1 + 4.50T + 17T^{2} \)
19 \( 1 - 6.73iT - 19T^{2} \)
29 \( 1 + 9.24iT - 29T^{2} \)
31 \( 1 - 0.592iT - 31T^{2} \)
37 \( 1 + 8.23T + 37T^{2} \)
41 \( 1 - 6.44T + 41T^{2} \)
43 \( 1 + 5.11T + 43T^{2} \)
47 \( 1 + 2.58T + 47T^{2} \)
53 \( 1 - 1.00iT - 53T^{2} \)
59 \( 1 + 0.599T + 59T^{2} \)
61 \( 1 + 6.29iT - 61T^{2} \)
67 \( 1 + 0.526T + 67T^{2} \)
71 \( 1 + 1.86iT - 71T^{2} \)
73 \( 1 - 9.34iT - 73T^{2} \)
79 \( 1 - 1.25T + 79T^{2} \)
83 \( 1 + 11.0T + 83T^{2} \)
89 \( 1 + 3.79T + 89T^{2} \)
97 \( 1 + 2.14iT - 97T^{2} \)
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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

−10.73043192207660263130910269829, −10.11668044621392011721694380150, −9.744646979446037706933236367828, −7.85129628469602358437362851160, −6.96381681932576988818126219352, −6.27554956802020706666691415411, −5.33896641934297270985873842075, −4.20367892848128214176888641841, −2.14026669410574435975753370052, −1.54526339670181378802748391092, 1.63745797618162007873238832418, 2.82383610713710024731591953522, 4.97711383365327071270585020302, 5.59349208696769222275683787405, 6.35430491568380921826973145294, 6.92807928478217261991598769746, 8.675347529029794600985510146112, 9.165947636251288286736931909305, 10.52896569646446892798003414729, 11.05562974514670455376249964693

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