Properties

Label 2-483-161.10-c1-0-17
Degree $2$
Conductor $483$
Sign $0.987 - 0.159i$
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  + (0.310 − 0.896i)2-s + (0.458 + 0.888i)3-s + (0.864 + 0.680i)4-s + (2.14 − 0.204i)5-s + (0.938 − 0.134i)6-s + (−1.73 + 1.99i)7-s + (2.47 − 1.58i)8-s + (−0.580 + 0.814i)9-s + (0.481 − 1.98i)10-s + (0.899 − 0.311i)11-s + (−0.208 + 1.08i)12-s + (0.327 + 1.11i)13-s + (1.25 + 2.17i)14-s + (1.16 + 1.81i)15-s + (−0.138 − 0.571i)16-s + (−2.90 − 1.16i)17-s + ⋯
L(s)  = 1  + (0.219 − 0.633i)2-s + (0.264 + 0.513i)3-s + (0.432 + 0.340i)4-s + (0.959 − 0.0916i)5-s + (0.383 − 0.0551i)6-s + (−0.655 + 0.755i)7-s + (0.874 − 0.562i)8-s + (−0.193 + 0.271i)9-s + (0.152 − 0.628i)10-s + (0.271 − 0.0938i)11-s + (−0.0601 + 0.311i)12-s + (0.0908 + 0.309i)13-s + (0.334 + 0.581i)14-s + (0.300 + 0.468i)15-s + (−0.0346 − 0.142i)16-s + (−0.704 − 0.281i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.987 - 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.987 - 0.159i$
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: \((2,\ 483,\ (\ :1/2),\ 0.987 - 0.159i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.18430 + 0.175811i\)
\(L(\frac12)\) \(\approx\) \(2.18430 + 0.175811i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.73 - 1.99i)T \)
23 \( 1 + (-1.08 + 4.67i)T \)
good2 \( 1 + (-0.310 + 0.896i)T + (-1.57 - 1.23i)T^{2} \)
5 \( 1 + (-2.14 + 0.204i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-0.899 + 0.311i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.327 - 1.11i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (2.90 + 1.16i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-0.161 + 0.0647i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-1.21 - 8.45i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (0.772 - 0.0367i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (3.63 + 2.59i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (3.22 + 1.47i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-7.02 + 10.9i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-8.45 + 4.88i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.40 - 5.66i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (6.24 + 1.51i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (8.62 + 4.44i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (1.20 + 6.22i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (4.69 + 5.41i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (7.64 - 9.72i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-2.62 + 2.74i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-2.15 - 4.70i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.459 + 9.63i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (4.50 - 9.87i)T + (-63.5 - 73.3i)T^{2} \)
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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

−10.81649192112124486971209798424, −10.32616107197640270277671260759, −9.168427771126001995810828233986, −8.801139439467884553663487572185, −7.20416930672309858106556712131, −6.32917548511202660170996082255, −5.24070414763817347402209656734, −3.95759622213343951294643751692, −2.84669623307072819894313080544, −1.96973456390719231155016022200, 1.45354912506861951530820411238, 2.72390164861924970674850387441, 4.30313812118445471428610968395, 5.77171494669795838226983954398, 6.28386806213926366394749176506, 7.13953509507458180044494576234, 7.921893874628278358041432698604, 9.263195134378890018780159915551, 10.03510992026055869435486853168, 10.83175358988687025120693619949

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