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Properties

Label 2-483-161.10-c1-0-12
Degree $2$
Conductor $483$
Sign $-0.816 - 0.577i$
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  + (−0.797 + 2.30i)2-s + (0.458 + 0.888i)3-s + (−3.09 − 2.43i)4-s + (1.48 − 0.142i)5-s + (−2.41 + 0.346i)6-s + (2.64 + 0.114i)7-s + (3.98 − 2.56i)8-s + (−0.580 + 0.814i)9-s + (−0.859 + 3.54i)10-s + (5.25 − 1.81i)11-s + (0.746 − 3.87i)12-s + (1.14 + 3.89i)13-s + (−2.37 + 5.99i)14-s + (0.808 + 1.25i)15-s + (0.863 + 3.56i)16-s + (2.36 + 0.945i)17-s + ⋯
L(s)  = 1  + (−0.563 + 1.62i)2-s + (0.264 + 0.513i)3-s + (−1.54 − 1.21i)4-s + (0.665 − 0.0635i)5-s + (−0.985 + 0.141i)6-s + (0.999 + 0.0433i)7-s + (1.40 − 0.905i)8-s + (−0.193 + 0.271i)9-s + (−0.271 + 1.12i)10-s + (1.58 − 0.548i)11-s + (0.215 − 1.11i)12-s + (0.317 + 1.07i)13-s + (−0.633 + 1.60i)14-s + (0.208 + 0.324i)15-s + (0.215 + 0.890i)16-s + (0.572 + 0.229i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.816 - 0.577i$
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: \((2,\ 483,\ (\ :1/2),\ -0.816 - 0.577i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.420805 + 1.32451i\)
\(L(\frac12)\) \(\approx\) \(0.420805 + 1.32451i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 - 0.114i)T \)
23 \( 1 + (-0.501 + 4.76i)T \)
good2 \( 1 + (0.797 - 2.30i)T + (-1.57 - 1.23i)T^{2} \)
5 \( 1 + (-1.48 + 0.142i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-5.25 + 1.81i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-1.14 - 3.89i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-2.36 - 0.945i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (4.40 - 1.76i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (0.100 + 0.701i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-1.67 + 0.0797i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (0.543 + 0.387i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (5.32 + 2.43i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (4.30 - 6.70i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-1.90 + 1.10i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.07 + 1.12i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (0.288 + 0.0700i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (10.0 + 5.18i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (2.58 + 13.4i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (2.97 + 3.43i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-2.80 + 3.56i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (8.36 - 8.77i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-2.45 - 5.38i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.806 - 16.9i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-4.75 + 10.4i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.11128789901034643375080733162, −10.02706452013603033948905400315, −9.143899229109399752833340186982, −8.662765555297787799606383787189, −7.88395420588078888708329432896, −6.55747973664589254608312156132, −6.10356577782973636933295095991, −4.91289349592141071006086108341, −4.00434244984890029039672725751, −1.63762037668329498201020355430, 1.24107773932101678758243742793, 1.98954480831316970178439700567, 3.29791437907042080185779227411, 4.42539589668325955606920905574, 5.86806457186866222648705506159, 7.23597539292603497943796268074, 8.344276629025128779080337117384, 8.995217199722636336395031376871, 9.888384064765295608831394102291, 10.59737858445064806139069262662

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