Properties

Label 2-483-161.10-c1-0-30
Degree $2$
Conductor $483$
Sign $-0.368 - 0.929i$
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  + (0.772 − 2.23i)2-s + (−0.458 − 0.888i)3-s + (−2.81 − 2.21i)4-s + (−0.0722 + 0.00690i)5-s + (−2.33 + 0.336i)6-s + (−2.49 − 0.873i)7-s + (−3.13 + 2.01i)8-s + (−0.580 + 0.814i)9-s + (−0.0404 + 0.166i)10-s + (−2.84 + 0.984i)11-s + (−0.677 + 3.51i)12-s + (0.146 + 0.498i)13-s + (−3.87 + 4.89i)14-s + (0.0392 + 0.0610i)15-s + (0.387 + 1.59i)16-s + (−2.14 − 0.857i)17-s + ⋯
L(s)  = 1  + (0.546 − 1.57i)2-s + (−0.264 − 0.513i)3-s + (−1.40 − 1.10i)4-s + (−0.0323 + 0.00308i)5-s + (−0.954 + 0.137i)6-s + (−0.943 − 0.330i)7-s + (−1.10 + 0.712i)8-s + (−0.193 + 0.271i)9-s + (−0.0127 + 0.0526i)10-s + (−0.857 + 0.296i)11-s + (−0.195 + 1.01i)12-s + (0.0406 + 0.138i)13-s + (−1.03 + 1.30i)14-s + (0.0101 + 0.0157i)15-s + (0.0969 + 0.399i)16-s + (−0.519 − 0.207i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.368 - 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.368 - 0.929i$
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: \((2,\ 483,\ (\ :1/2),\ -0.368 - 0.929i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.506468 + 0.745417i\)
\(L(\frac12)\) \(\approx\) \(0.506468 + 0.745417i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.49 + 0.873i)T \)
23 \( 1 + (-1.82 + 4.43i)T \)
good2 \( 1 + (-0.772 + 2.23i)T + (-1.57 - 1.23i)T^{2} \)
5 \( 1 + (0.0722 - 0.00690i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (2.84 - 0.984i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.146 - 0.498i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (2.14 + 0.857i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-2.45 + 0.981i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (0.422 + 2.93i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-5.95 + 0.283i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (0.156 + 0.111i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (6.38 + 2.91i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (0.950 - 1.47i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-10.8 + 6.27i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.14 + 6.44i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (-11.5 - 2.80i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (7.81 + 4.03i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-0.126 - 0.654i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (6.65 + 7.67i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-3.53 + 4.49i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (2.46 - 2.58i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (3.21 + 7.04i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.152 + 3.20i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (2.85 - 6.24i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.44222705818404522872509398461, −9.962780044441746236970262623411, −8.925117513970184269892179448556, −7.56689197760600934028294691206, −6.53140862926941531546067602945, −5.31941853845640553172418882574, −4.29033501671643374740115630481, −3.11179846633735746857571966296, −2.15850988104822716733502723883, −0.45680744150034043535833279201, 3.07629178404149825882508425653, 4.20341382695922398893803152834, 5.35206292178120896310025400811, 5.90010574123542905963275480321, 6.83815140790927652726964153588, 7.79034262299286367854001067806, 8.703441123191379081583038757398, 9.610780783619288168882579368606, 10.55408154489553455322412646886, 11.73851880185594740111968078633

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