Properties

Label 2-483-161.4-c1-0-16
Degree $2$
Conductor $483$
Sign $0.999 - 0.0228i$
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  + (−0.851 + 1.19i)2-s + (−0.235 + 0.971i)3-s + (−0.0511 − 0.147i)4-s + (0.117 − 2.46i)5-s + (−0.961 − 1.10i)6-s + (−2.08 − 1.62i)7-s + (−2.59 − 0.762i)8-s + (−0.888 − 0.458i)9-s + (2.84 + 2.23i)10-s + (2.66 + 3.74i)11-s + (0.155 − 0.0148i)12-s + (0.969 − 6.74i)13-s + (3.72 − 1.10i)14-s + (2.36 + 0.693i)15-s + (3.37 − 2.65i)16-s + (0.693 − 0.133i)17-s + ⋯
L(s)  = 1  + (−0.602 + 0.845i)2-s + (−0.136 + 0.561i)3-s + (−0.0255 − 0.0738i)4-s + (0.0524 − 1.10i)5-s + (−0.392 − 0.453i)6-s + (−0.787 − 0.615i)7-s + (−0.918 − 0.269i)8-s + (−0.296 − 0.152i)9-s + (0.898 + 0.706i)10-s + (0.803 + 1.12i)11-s + (0.0448 − 0.00428i)12-s + (0.268 − 1.86i)13-s + (0.995 − 0.295i)14-s + (0.610 + 0.179i)15-s + (0.842 − 0.662i)16-s + (0.168 − 0.0324i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0228i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0228i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.999 - 0.0228i$
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: \((2,\ 483,\ (\ :1/2),\ 0.999 - 0.0228i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.824857 + 0.00943310i\)
\(L(\frac12)\) \(\approx\) \(0.824857 + 0.00943310i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.08 + 1.62i)T \)
23 \( 1 + (3.35 + 3.42i)T \)
good2 \( 1 + (0.851 - 1.19i)T + (-0.654 - 1.89i)T^{2} \)
5 \( 1 + (-0.117 + 2.46i)T + (-4.97 - 0.475i)T^{2} \)
11 \( 1 + (-2.66 - 3.74i)T + (-3.59 + 10.3i)T^{2} \)
13 \( 1 + (-0.969 + 6.74i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (-0.693 + 0.133i)T + (15.7 - 6.31i)T^{2} \)
19 \( 1 + (-5.51 - 1.06i)T + (17.6 + 7.06i)T^{2} \)
29 \( 1 + (2.24 + 2.58i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-5.05 + 4.81i)T + (1.47 - 30.9i)T^{2} \)
37 \( 1 + (2.31 + 1.19i)T + (21.4 + 30.1i)T^{2} \)
41 \( 1 + (1.97 + 1.27i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (-4.40 + 1.29i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + (-5.59 - 9.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.306 - 0.122i)T + (38.3 + 36.5i)T^{2} \)
59 \( 1 + (10.8 + 8.54i)T + (13.9 + 57.3i)T^{2} \)
61 \( 1 + (2.74 + 11.3i)T + (-54.2 + 27.9i)T^{2} \)
67 \( 1 + (0.775 + 0.0740i)T + (65.7 + 12.6i)T^{2} \)
71 \( 1 + (0.752 + 1.64i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (0.649 + 1.87i)T + (-57.3 + 45.1i)T^{2} \)
79 \( 1 + (-1.34 + 0.538i)T + (57.1 - 54.5i)T^{2} \)
83 \( 1 + (-1.86 + 1.19i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-6.44 - 6.14i)T + (4.23 + 88.8i)T^{2} \)
97 \( 1 + (8.09 + 5.20i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.65479395821074219010627683243, −9.617770337835774978943380110775, −9.407454277323010075841444413237, −8.146726522617981968919225275454, −7.57352465846998426811571075699, −6.37753358129598416855482716116, −5.52192555314184249237142014391, −4.30547295933542249127520362381, −3.19656668233839217048618831049, −0.67513290243393649157584404792, 1.43136811907260897403353068602, 2.73421231257885420359522929049, 3.56385832695050540943711318617, 5.72445199358539836600459142807, 6.41997939496203870912048867078, 7.12399689749972249905740969778, 8.731739541476726129243318304383, 9.226820711980337085274478535374, 10.15665347787512852404475690878, 11.09454379398958424653629084340

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