Properties

Label 2-483-161.145-c1-0-3
Degree $2$
Conductor $483$
Sign $-0.933 + 0.358i$
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  + (0.679 + 1.96i)2-s + (0.458 − 0.888i)3-s + (−1.81 + 1.42i)4-s + (−3.97 − 0.379i)5-s + (2.05 + 0.295i)6-s + (0.921 + 2.48i)7-s + (−0.546 − 0.351i)8-s + (−0.580 − 0.814i)9-s + (−1.95 − 8.05i)10-s + (−3.49 − 1.20i)11-s + (0.437 + 2.27i)12-s + (−1.42 + 4.84i)13-s + (−4.24 + 3.49i)14-s + (−2.15 + 3.35i)15-s + (−0.772 + 3.18i)16-s + (−3.63 + 1.45i)17-s + ⋯
L(s)  = 1  + (0.480 + 1.38i)2-s + (0.264 − 0.513i)3-s + (−0.908 + 0.714i)4-s + (−1.77 − 0.169i)5-s + (0.839 + 0.120i)6-s + (0.348 + 0.937i)7-s + (−0.193 − 0.124i)8-s + (−0.193 − 0.271i)9-s + (−0.617 − 2.54i)10-s + (−1.05 − 0.364i)11-s + (0.126 + 0.655i)12-s + (−0.394 + 1.34i)13-s + (−1.13 + 0.933i)14-s + (−0.556 + 0.866i)15-s + (−0.193 + 0.796i)16-s + (−0.881 + 0.353i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.933 + 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.933 + 0.358i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.933 + 0.358i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.147229 - 0.794664i\)
\(L(\frac12)\) \(\approx\) \(0.147229 - 0.794664i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.921 - 2.48i)T \)
23 \( 1 + (-4.30 - 2.11i)T \)
good2 \( 1 + (-0.679 - 1.96i)T + (-1.57 + 1.23i)T^{2} \)
5 \( 1 + (3.97 + 0.379i)T + (4.90 + 0.946i)T^{2} \)
11 \( 1 + (3.49 + 1.20i)T + (8.64 + 6.79i)T^{2} \)
13 \( 1 + (1.42 - 4.84i)T + (-10.9 - 7.02i)T^{2} \)
17 \( 1 + (3.63 - 1.45i)T + (12.3 - 11.7i)T^{2} \)
19 \( 1 + (6.01 + 2.40i)T + (13.7 + 13.1i)T^{2} \)
29 \( 1 + (-0.765 + 5.32i)T + (-27.8 - 8.17i)T^{2} \)
31 \( 1 + (-2.45 - 0.116i)T + (30.8 + 2.94i)T^{2} \)
37 \( 1 + (-0.999 + 0.711i)T + (12.1 - 34.9i)T^{2} \)
41 \( 1 + (-1.81 + 0.830i)T + (26.8 - 30.9i)T^{2} \)
43 \( 1 + (-5.70 - 8.87i)T + (-17.8 + 39.1i)T^{2} \)
47 \( 1 + (-3.43 - 1.98i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-1.54 + 1.62i)T + (-2.52 - 52.9i)T^{2} \)
59 \( 1 + (6.82 - 1.65i)T + (52.4 - 27.0i)T^{2} \)
61 \( 1 + (-3.80 + 1.96i)T + (35.3 - 49.6i)T^{2} \)
67 \( 1 + (-0.334 + 1.73i)T + (-62.2 - 24.9i)T^{2} \)
71 \( 1 + (8.56 - 9.88i)T + (-10.1 - 70.2i)T^{2} \)
73 \( 1 + (-7.91 - 10.0i)T + (-17.2 + 70.9i)T^{2} \)
79 \( 1 + (2.88 + 3.03i)T + (-3.75 + 78.9i)T^{2} \)
83 \( 1 + (1.63 - 3.57i)T + (-54.3 - 62.7i)T^{2} \)
89 \( 1 + (0.403 + 8.47i)T + (-88.5 + 8.45i)T^{2} \)
97 \( 1 + (-1.21 - 2.66i)T + (-63.5 + 73.3i)T^{2} \)
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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

−11.48583073831716687481612006742, −11.04817034520192864021556368288, −8.974116282884531250473809040944, −8.438194166543183849619881962257, −7.76230830221092872500969317970, −6.99979552168641711181435700612, −6.10876179776683262827042257590, −4.78701512615799384354402755175, −4.21436500325569975080952243460, −2.49919286376956776396876662545, 0.39266618475749447714752596051, 2.59727938889375034622227178431, 3.50513563572434513746381591671, 4.38743621651206586539263284311, 4.93339587566317463672844869390, 7.15207049569203169016586794030, 7.80913308195210959043487178791, 8.717791775661332835319271939202, 10.26697326788679592924203776709, 10.71339568365979410922433696434

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