Properties

Label 2-483-161.19-c1-0-10
Degree $2$
Conductor $483$
Sign $0.470 + 0.882i$
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  + (−1.83 − 0.175i)2-s + (−0.690 + 0.723i)3-s + (1.38 + 0.267i)4-s + (−1.78 + 0.917i)5-s + (1.39 − 1.20i)6-s + (−2.33 − 1.23i)7-s + (1.04 + 0.306i)8-s + (−0.0475 − 0.998i)9-s + (3.43 − 1.37i)10-s + (0.533 + 5.58i)11-s + (−1.14 + 0.818i)12-s + (−3.84 − 0.553i)13-s + (4.08 + 2.68i)14-s + (0.564 − 1.92i)15-s + (−4.48 − 1.79i)16-s + (−0.0974 + 0.281i)17-s + ⋯
L(s)  = 1  + (−1.30 − 0.124i)2-s + (−0.398 + 0.417i)3-s + (0.692 + 0.133i)4-s + (−0.796 + 0.410i)5-s + (0.569 − 0.493i)6-s + (−0.883 − 0.467i)7-s + (0.368 + 0.108i)8-s + (−0.0158 − 0.332i)9-s + (1.08 − 0.434i)10-s + (0.160 + 1.68i)11-s + (−0.331 + 0.236i)12-s + (−1.06 − 0.153i)13-s + (1.09 + 0.718i)14-s + (0.145 − 0.496i)15-s + (−1.12 − 0.448i)16-s + (−0.0236 + 0.0683i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.470 + 0.882i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.470 + 0.882i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.210134 - 0.126078i\)
\(L(\frac12)\) \(\approx\) \(0.210134 - 0.126078i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.690 - 0.723i)T \)
7 \( 1 + (2.33 + 1.23i)T \)
23 \( 1 + (-2.41 + 4.14i)T \)
good2 \( 1 + (1.83 + 0.175i)T + (1.96 + 0.378i)T^{2} \)
5 \( 1 + (1.78 - 0.917i)T + (2.90 - 4.07i)T^{2} \)
11 \( 1 + (-0.533 - 5.58i)T + (-10.8 + 2.08i)T^{2} \)
13 \( 1 + (3.84 + 0.553i)T + (12.4 + 3.66i)T^{2} \)
17 \( 1 + (0.0974 - 0.281i)T + (-13.3 - 10.5i)T^{2} \)
19 \( 1 + (0.874 + 2.52i)T + (-14.9 + 11.7i)T^{2} \)
29 \( 1 + (-0.420 - 0.484i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-4.06 + 0.986i)T + (27.5 - 14.2i)T^{2} \)
37 \( 1 + (-11.4 + 0.544i)T + (36.8 - 3.51i)T^{2} \)
41 \( 1 + (-2.97 + 4.62i)T + (-17.0 - 37.2i)T^{2} \)
43 \( 1 + (3.58 + 12.2i)T + (-36.1 + 23.2i)T^{2} \)
47 \( 1 + (5.76 + 3.32i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.24 + 4.13i)T + (-12.4 + 51.5i)T^{2} \)
59 \( 1 + (0.264 + 0.660i)T + (-42.7 + 40.7i)T^{2} \)
61 \( 1 + (-9.28 + 8.85i)T + (2.90 - 60.9i)T^{2} \)
67 \( 1 + (-8.35 - 5.94i)T + (21.9 + 63.3i)T^{2} \)
71 \( 1 + (-0.457 - 1.00i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-2.56 + 13.3i)T + (-67.7 - 27.1i)T^{2} \)
79 \( 1 + (6.90 - 8.77i)T + (-18.6 - 76.7i)T^{2} \)
83 \( 1 + (8.84 - 5.68i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-0.297 + 1.22i)T + (-79.1 - 40.7i)T^{2} \)
97 \( 1 + (2.10 + 1.35i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.54156598887482441449049995011, −9.839150531127549784974284899183, −9.426789032206040706361151374645, −8.154870513495749132368227051631, −7.14692191926910940481536774107, −6.82309339471108583804171692040, −4.91259976819369502355396161277, −4.04895300855962466319063105594, −2.44250121995398759366479715546, −0.32277344615339673292904880482, 0.941931738793622525810298550011, 2.90991552955182256877434804822, 4.41463315079376372120140019944, 5.85155448021326849097343377736, 6.72245344495909528529222111399, 7.87362265203263963742230049512, 8.296798565617513197778152210290, 9.345622711670525206745281275353, 9.958900531010470497615594101694, 11.23064978734318337649363459494

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