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Properties

Label 2-483-69.68-c1-0-23
Degree $2$
Conductor $483$
Sign $0.897 - 0.441i$
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  − 0.369i·2-s + (0.867 + 1.49i)3-s + 1.86·4-s − 1.03·5-s + (0.554 − 0.320i)6-s − i·7-s − 1.42i·8-s + (−1.49 + 2.60i)9-s + 0.381i·10-s + 0.430·11-s + (1.61 + 2.79i)12-s + 4.27·13-s − 0.369·14-s + (−0.895 − 1.54i)15-s + 3.19·16-s + 7.25·17-s + ⋯
L(s)  = 1  − 0.261i·2-s + (0.500 + 0.865i)3-s + 0.931·4-s − 0.461·5-s + (0.226 − 0.130i)6-s − 0.377i·7-s − 0.504i·8-s + (−0.498 + 0.866i)9-s + 0.120i·10-s + 0.129·11-s + (0.466 + 0.806i)12-s + 1.18·13-s − 0.0987·14-s + (−0.231 − 0.399i)15-s + 0.799·16-s + 1.75·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.897 - 0.441i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.897 - 0.441i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.897 - 0.441i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.92780 + 0.448621i\)
\(L(\frac12)\) \(\approx\) \(1.92780 + 0.448621i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.867 - 1.49i)T \)
7 \( 1 + iT \)
23 \( 1 + (4.78 + 0.321i)T \)
good2 \( 1 + 0.369iT - 2T^{2} \)
5 \( 1 + 1.03T + 5T^{2} \)
11 \( 1 - 0.430T + 11T^{2} \)
13 \( 1 - 4.27T + 13T^{2} \)
17 \( 1 - 7.25T + 17T^{2} \)
19 \( 1 - 3.53iT - 19T^{2} \)
29 \( 1 - 4.52iT - 29T^{2} \)
31 \( 1 + 3.23T + 31T^{2} \)
37 \( 1 - 8.22iT - 37T^{2} \)
41 \( 1 + 4.85iT - 41T^{2} \)
43 \( 1 + 9.90iT - 43T^{2} \)
47 \( 1 + 0.312iT - 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 + 8.46iT - 59T^{2} \)
61 \( 1 + 11.8iT - 61T^{2} \)
67 \( 1 + 5.50iT - 67T^{2} \)
71 \( 1 + 12.5iT - 71T^{2} \)
73 \( 1 + 4.83T + 73T^{2} \)
79 \( 1 - 17.2iT - 79T^{2} \)
83 \( 1 - 10.7T + 83T^{2} \)
89 \( 1 - 4.36T + 89T^{2} \)
97 \( 1 - 4.63iT - 97T^{2} \)
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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

−10.87322720856764381812012155290, −10.30550044698311679803843242870, −9.501343602944300677736509871803, −8.149212034506522385082309214831, −7.74139156340489728631596061040, −6.36814188984852902671178252125, −5.35663301284854156199286993950, −3.69744516547076024483326943872, −3.46099870738988631898592723584, −1.70753060696453066544896145013, 1.41579307779124018092319281243, 2.75187293605108719694838604498, 3.78236785221669010353018103717, 5.76622080477071335345063218797, 6.25011316298649649213292833363, 7.52903191162504721613827519451, 7.86604791310157054876219015164, 8.858170132576418900798716113669, 9.990371343057969038729689275379, 11.28015702902507735131772876337

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