Properties

Label 2-483-161.10-c1-0-2
Degree $2$
Conductor $483$
Sign $-0.893 + 0.448i$
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  + (−0.473 + 1.36i)2-s + (−0.458 − 0.888i)3-s + (−0.0749 − 0.0589i)4-s + (−1.20 + 0.115i)5-s + (1.43 − 0.206i)6-s + (0.607 − 2.57i)7-s + (−2.31 + 1.49i)8-s + (−0.580 + 0.814i)9-s + (0.413 − 1.70i)10-s + (−3.29 + 1.14i)11-s + (−0.0180 + 0.0936i)12-s + (0.982 + 3.34i)13-s + (3.23 + 2.05i)14-s + (0.655 + 1.02i)15-s + (−0.985 − 4.06i)16-s + (−3.34 − 1.33i)17-s + ⋯
L(s)  = 1  + (−0.334 + 0.967i)2-s + (−0.264 − 0.513i)3-s + (−0.0374 − 0.0294i)4-s + (−0.539 + 0.0515i)5-s + (0.584 − 0.0841i)6-s + (0.229 − 0.973i)7-s + (−0.820 + 0.526i)8-s + (−0.193 + 0.271i)9-s + (0.130 − 0.539i)10-s + (−0.994 + 0.344i)11-s + (−0.00520 + 0.0270i)12-s + (0.272 + 0.927i)13-s + (0.864 + 0.548i)14-s + (0.169 + 0.263i)15-s + (−0.246 − 1.01i)16-s + (−0.811 − 0.324i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 + 0.448i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.893 + 0.448i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.893 + 0.448i$
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: \((2,\ 483,\ (\ :1/2),\ -0.893 + 0.448i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0511717 - 0.216280i\)
\(L(\frac12)\) \(\approx\) \(0.0511717 - 0.216280i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.607 + 2.57i)T \)
23 \( 1 + (1.89 - 4.40i)T \)
good2 \( 1 + (0.473 - 1.36i)T + (-1.57 - 1.23i)T^{2} \)
5 \( 1 + (1.20 - 0.115i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (3.29 - 1.14i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.982 - 3.34i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (3.34 + 1.33i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (5.08 - 2.03i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (0.311 + 2.16i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-8.47 + 0.403i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (6.95 + 4.95i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (1.67 + 0.764i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (6.27 - 9.76i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-6.20 + 3.58i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (5.62 + 5.90i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (4.76 + 1.15i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (-7.10 - 3.66i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (1.55 + 8.08i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-10.7 - 12.4i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-7.51 + 9.55i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-0.771 + 0.809i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-3.80 - 8.33i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.375 - 7.88i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (5.32 - 11.6i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.47654466118311239376395680674, −10.73025099094829809825419554256, −9.568126648163597562725411935781, −8.242671186434646656543052885376, −7.87912002991001989791435993652, −6.93864762998753001437013734021, −6.37310050782615533660784709164, −5.05010812288603268985403319121, −3.87424189955553586037918379738, −2.14508062492186252956167623011, 0.14119908302388306967912861411, 2.23514449224288884384576254803, 3.19734808698592897674329728984, 4.54043132305617959570969510086, 5.68516534616891839880549353193, 6.57757517687793301369206771640, 8.336072187313102880187256998247, 8.607743434298650569112271324971, 9.912129056604813379857481466429, 10.64876717431990849569310685908

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