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Properties

Label 2-483-161.19-c1-0-19
Degree $2$
Conductor $483$
Sign $-0.297 + 0.954i$
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  + (−2.78 − 0.265i)2-s + (0.690 − 0.723i)3-s + (5.70 + 1.10i)4-s + (0.0901 − 0.0464i)5-s + (−2.11 + 1.83i)6-s + (2.63 − 0.236i)7-s + (−10.2 − 3.00i)8-s + (−0.0475 − 0.998i)9-s + (−0.263 + 0.105i)10-s + (−0.291 − 3.04i)11-s + (4.73 − 3.37i)12-s + (−5.20 − 0.749i)13-s + (−7.39 − 0.0431i)14-s + (0.0285 − 0.0972i)15-s + (16.8 + 6.74i)16-s + (0.524 − 1.51i)17-s + ⋯
L(s)  = 1  + (−1.96 − 0.187i)2-s + (0.398 − 0.417i)3-s + (2.85 + 0.550i)4-s + (0.0402 − 0.0207i)5-s + (−0.862 + 0.747i)6-s + (0.996 − 0.0892i)7-s + (−3.61 − 1.06i)8-s + (−0.0158 − 0.332i)9-s + (−0.0831 + 0.0333i)10-s + (−0.0877 − 0.919i)11-s + (1.36 − 0.973i)12-s + (−1.44 − 0.207i)13-s + (−1.97 − 0.0115i)14-s + (0.00737 − 0.0251i)15-s + (4.21 + 1.68i)16-s + (0.127 − 0.367i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.297 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.297 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.297 + 0.954i$
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: \((2,\ 483,\ (\ :1/2),\ -0.297 + 0.954i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.369512 - 0.502382i\)
\(L(\frac12)\) \(\approx\) \(0.369512 - 0.502382i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 + 0.236i)T \)
23 \( 1 + (-4.54 - 1.53i)T \)
good2 \( 1 + (2.78 + 0.265i)T + (1.96 + 0.378i)T^{2} \)
5 \( 1 + (-0.0901 + 0.0464i)T + (2.90 - 4.07i)T^{2} \)
11 \( 1 + (0.291 + 3.04i)T + (-10.8 + 2.08i)T^{2} \)
13 \( 1 + (5.20 + 0.749i)T + (12.4 + 3.66i)T^{2} \)
17 \( 1 + (-0.524 + 1.51i)T + (-13.3 - 10.5i)T^{2} \)
19 \( 1 + (2.69 + 7.78i)T + (-14.9 + 11.7i)T^{2} \)
29 \( 1 + (-2.32 - 2.67i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-4.80 + 1.16i)T + (27.5 - 14.2i)T^{2} \)
37 \( 1 + (2.77 - 0.132i)T + (36.8 - 3.51i)T^{2} \)
41 \( 1 + (-1.21 + 1.88i)T + (-17.0 - 37.2i)T^{2} \)
43 \( 1 + (2.33 + 7.94i)T + (-36.1 + 23.2i)T^{2} \)
47 \( 1 + (-1.46 - 0.845i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.97 + 5.05i)T + (-12.4 + 51.5i)T^{2} \)
59 \( 1 + (2.85 + 7.12i)T + (-42.7 + 40.7i)T^{2} \)
61 \( 1 + (-5.20 + 4.96i)T + (2.90 - 60.9i)T^{2} \)
67 \( 1 + (2.83 + 2.02i)T + (21.9 + 63.3i)T^{2} \)
71 \( 1 + (-2.38 - 5.22i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (1.33 - 6.95i)T + (-67.7 - 27.1i)T^{2} \)
79 \( 1 + (-2.21 + 2.82i)T + (-18.6 - 76.7i)T^{2} \)
83 \( 1 + (1.82 - 1.17i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (0.298 - 1.23i)T + (-79.1 - 40.7i)T^{2} \)
97 \( 1 + (3.47 + 2.23i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.65062569288815460045674456175, −9.597905490910358668113765575727, −8.870852063127431757259474582129, −8.193409499497860084275610474620, −7.35056133414719609302989180178, −6.77286628940458329363162434834, −5.23436414180836820045336118085, −3.03989808489241581253106739517, −2.10710315781660920944255610892, −0.66520892771403532288609595128, 1.67866980181231331260644884944, 2.60256664183278287038842459724, 4.61487525649586778043271755407, 6.01004981614012018069267138156, 7.18599303800133494273172537171, 7.940669824998943829592995898044, 8.467348956432874397105443931041, 9.558049775601576986981320986638, 10.12358887314646899616499603282, 10.71285346951854409759769128560

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