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Properties

Label 2-483-161.4-c1-0-11
Degree $2$
Conductor $483$
Sign $0.657 - 0.753i$
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  + (−1.32 + 1.86i)2-s + (0.235 − 0.971i)3-s + (−1.05 − 3.04i)4-s + (−0.0837 + 1.75i)5-s + (1.49 + 1.72i)6-s + (0.00745 − 2.64i)7-s + (2.67 + 0.786i)8-s + (−0.888 − 0.458i)9-s + (−3.16 − 2.48i)10-s + (0.920 + 1.29i)11-s + (−3.20 + 0.306i)12-s + (−0.0452 + 0.314i)13-s + (4.91 + 3.52i)14-s + (1.68 + 0.495i)15-s + (0.0500 − 0.0393i)16-s + (4.22 − 0.815i)17-s + ⋯
L(s)  = 1  + (−0.937 + 1.31i)2-s + (0.136 − 0.561i)3-s + (−0.526 − 1.52i)4-s + (−0.0374 + 0.786i)5-s + (0.610 + 0.705i)6-s + (0.00281 − 0.999i)7-s + (0.947 + 0.278i)8-s + (−0.296 − 0.152i)9-s + (−0.999 − 0.786i)10-s + (0.277 + 0.389i)11-s + (−0.925 + 0.0884i)12-s + (−0.0125 + 0.0873i)13-s + (1.31 + 0.940i)14-s + (0.436 + 0.128i)15-s + (0.0125 − 0.00984i)16-s + (1.02 − 0.197i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.657 - 0.753i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.657 - 0.753i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.807893 + 0.366939i\)
\(L(\frac12)\) \(\approx\) \(0.807893 + 0.366939i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.235 + 0.971i)T \)
7 \( 1 + (-0.00745 + 2.64i)T \)
23 \( 1 + (-2.81 + 3.88i)T \)
good2 \( 1 + (1.32 - 1.86i)T + (-0.654 - 1.89i)T^{2} \)
5 \( 1 + (0.0837 - 1.75i)T + (-4.97 - 0.475i)T^{2} \)
11 \( 1 + (-0.920 - 1.29i)T + (-3.59 + 10.3i)T^{2} \)
13 \( 1 + (0.0452 - 0.314i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-4.22 + 0.815i)T + (15.7 - 6.31i)T^{2} \)
19 \( 1 + (0.509 + 0.0982i)T + (17.6 + 7.06i)T^{2} \)
29 \( 1 + (-1.27 - 1.47i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-5.23 + 4.98i)T + (1.47 - 30.9i)T^{2} \)
37 \( 1 + (-6.73 - 3.47i)T + (21.4 + 30.1i)T^{2} \)
41 \( 1 + (-4.89 - 3.14i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (-0.302 + 0.0889i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + (-1.27 - 2.21i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.76 + 2.70i)T + (38.3 + 36.5i)T^{2} \)
59 \( 1 + (-0.130 - 0.102i)T + (13.9 + 57.3i)T^{2} \)
61 \( 1 + (-2.29 - 9.45i)T + (-54.2 + 27.9i)T^{2} \)
67 \( 1 + (9.13 + 0.872i)T + (65.7 + 12.6i)T^{2} \)
71 \( 1 + (0.869 + 1.90i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (2.59 + 7.50i)T + (-57.3 + 45.1i)T^{2} \)
79 \( 1 + (-10.1 + 4.08i)T + (57.1 - 54.5i)T^{2} \)
83 \( 1 + (-8.25 + 5.30i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (2.39 + 2.28i)T + (4.23 + 88.8i)T^{2} \)
97 \( 1 + (-2.29 - 1.47i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.74904620661701623803734372146, −10.01111654110360957963781842117, −9.163589022856249400951101712153, −8.005991793873010263066288752987, −7.50005077246528961376898570172, −6.71959446347507204733335095255, −6.12705441510406772616390698189, −4.62917542010882509484046054206, −3.00972825346660243849473286196, −0.965251845093935585429521013452, 1.13506476919664157661548047288, 2.61435715563688508469801661327, 3.59919275931545059983335233080, 4.94890369339491303793840663475, 5.99389582544527959453299078853, 7.86009947192956748056557294929, 8.625490487447829414577173445671, 9.207987353451898655464487216296, 9.858845582513338595252345711924, 10.82590014580507294271058782794

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