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Properties

Label 2-483-161.10-c1-0-21
Degree $2$
Conductor $483$
Sign $0.995 - 0.0919i$
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  + (−0.131 + 0.379i)2-s + (−0.458 − 0.888i)3-s + (1.44 + 1.13i)4-s + (3.76 − 0.359i)5-s + (0.397 − 0.0570i)6-s + (2.39 − 1.11i)7-s + (−1.29 + 0.832i)8-s + (−0.580 + 0.814i)9-s + (−0.358 + 1.47i)10-s + (1.70 − 0.590i)11-s + (0.348 − 1.80i)12-s + (−0.0457 − 0.155i)13-s + (0.109 + 1.05i)14-s + (−2.04 − 3.18i)15-s + (0.721 + 2.97i)16-s + (−6.34 − 2.53i)17-s + ⋯
L(s)  = 1  + (−0.0927 + 0.268i)2-s + (−0.264 − 0.513i)3-s + (0.722 + 0.568i)4-s + (1.68 − 0.160i)5-s + (0.162 − 0.0233i)6-s + (0.906 − 0.422i)7-s + (−0.458 + 0.294i)8-s + (−0.193 + 0.271i)9-s + (−0.113 + 0.466i)10-s + (0.514 − 0.178i)11-s + (0.100 − 0.521i)12-s + (−0.0126 − 0.0432i)13-s + (0.0292 + 0.282i)14-s + (−0.528 − 0.822i)15-s + (0.180 + 0.743i)16-s + (−1.53 − 0.615i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.995 - 0.0919i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.995 - 0.0919i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.96621 + 0.0905990i\)
\(L(\frac12)\) \(\approx\) \(1.96621 + 0.0905990i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.458 + 0.888i)T \)
7 \( 1 + (-2.39 + 1.11i)T \)
23 \( 1 + (0.586 + 4.75i)T \)
good2 \( 1 + (0.131 - 0.379i)T + (-1.57 - 1.23i)T^{2} \)
5 \( 1 + (-3.76 + 0.359i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-1.70 + 0.590i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (0.0457 + 0.155i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (6.34 + 2.53i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (6.40 - 2.56i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-0.575 - 4.00i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (6.84 - 0.326i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-7.61 - 5.42i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (4.60 + 2.10i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (0.916 - 1.42i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-3.70 + 2.14i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.600 + 0.629i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (-6.68 - 1.62i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (-8.14 - 4.19i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-0.941 - 4.88i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (10.1 + 11.6i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (2.80 - 3.56i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (9.80 - 10.2i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (5.99 + 13.1i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.148 + 3.11i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (2.81 - 6.15i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.97011118691371267961956681897, −10.31955163871536436735468653080, −8.913830280894962115788737833191, −8.401394123160604583998824179151, −7.04200968252014027902793483875, −6.50843269484567179845259201783, −5.63945405634544896009675586267, −4.39563344898263081059215358529, −2.45559998257343362440718580650, −1.69335771404983552125204259229, 1.77571928232120406779490207411, 2.37724319488033180298227695033, 4.34551128411439414094260808130, 5.53658544243674788551467040015, 6.12483821792636137254178283807, 6.96044786783377759229769314350, 8.690083923647449151499038608790, 9.374082364257749628421114810312, 10.14533182997737250251659064008, 11.04914790256630122051746480401

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