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Properties

Label 2-483-161.19-c1-0-30
Degree $2$
Conductor $483$
Sign $0.946 + 0.322i$
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  + (2.32 + 0.222i)2-s + (0.690 − 0.723i)3-s + (3.39 + 0.653i)4-s + (0.830 − 0.428i)5-s + (1.76 − 1.52i)6-s + (−0.521 − 2.59i)7-s + (3.26 + 0.957i)8-s + (−0.0475 − 0.998i)9-s + (2.02 − 0.810i)10-s + (0.427 + 4.47i)11-s + (2.81 − 2.00i)12-s + (−4.42 − 0.636i)13-s + (−0.636 − 6.14i)14-s + (0.263 − 0.896i)15-s + (0.955 + 0.382i)16-s + (−0.782 + 2.26i)17-s + ⋯
L(s)  = 1  + (1.64 + 0.156i)2-s + (0.398 − 0.417i)3-s + (1.69 + 0.326i)4-s + (0.371 − 0.191i)5-s + (0.720 − 0.624i)6-s + (−0.197 − 0.980i)7-s + (1.15 + 0.338i)8-s + (−0.0158 − 0.332i)9-s + (0.640 − 0.256i)10-s + (0.128 + 1.34i)11-s + (0.812 − 0.578i)12-s + (−1.22 − 0.176i)13-s + (−0.170 − 1.64i)14-s + (0.0679 − 0.231i)15-s + (0.238 + 0.0956i)16-s + (−0.189 + 0.548i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.946 + 0.322i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.946 + 0.322i$
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: \((2,\ 483,\ (\ :1/2),\ 0.946 + 0.322i)\)

Particular Values

\(L(1)\) \(\approx\) \(3.75446 - 0.621165i\)
\(L(\frac12)\) \(\approx\) \(3.75446 - 0.621165i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.521 + 2.59i)T \)
23 \( 1 + (-4.29 - 2.13i)T \)
good2 \( 1 + (-2.32 - 0.222i)T + (1.96 + 0.378i)T^{2} \)
5 \( 1 + (-0.830 + 0.428i)T + (2.90 - 4.07i)T^{2} \)
11 \( 1 + (-0.427 - 4.47i)T + (-10.8 + 2.08i)T^{2} \)
13 \( 1 + (4.42 + 0.636i)T + (12.4 + 3.66i)T^{2} \)
17 \( 1 + (0.782 - 2.26i)T + (-13.3 - 10.5i)T^{2} \)
19 \( 1 + (-0.188 - 0.544i)T + (-14.9 + 11.7i)T^{2} \)
29 \( 1 + (-0.838 - 0.968i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-3.90 + 0.946i)T + (27.5 - 14.2i)T^{2} \)
37 \( 1 + (-1.78 + 0.0852i)T + (36.8 - 3.51i)T^{2} \)
41 \( 1 + (-2.83 + 4.41i)T + (-17.0 - 37.2i)T^{2} \)
43 \( 1 + (-1.67 - 5.69i)T + (-36.1 + 23.2i)T^{2} \)
47 \( 1 + (4.41 + 2.55i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.69 - 4.69i)T + (-12.4 + 51.5i)T^{2} \)
59 \( 1 + (2.06 + 5.15i)T + (-42.7 + 40.7i)T^{2} \)
61 \( 1 + (-2.85 + 2.72i)T + (2.90 - 60.9i)T^{2} \)
67 \( 1 + (6.05 + 4.31i)T + (21.9 + 63.3i)T^{2} \)
71 \( 1 + (6.42 + 14.0i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-2.01 + 10.4i)T + (-67.7 - 27.1i)T^{2} \)
79 \( 1 + (3.21 - 4.08i)T + (-18.6 - 76.7i)T^{2} \)
83 \( 1 + (2.13 - 1.36i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (1.81 - 7.48i)T + (-79.1 - 40.7i)T^{2} \)
97 \( 1 + (4.69 + 3.01i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.25675236312208454241329885407, −10.08817606354375557036259929620, −9.319354969573033057113340164374, −7.62856232726347216183757293551, −7.14913774224119430904124226744, −6.22837741045223366365004327889, −5.00450991975601995368973180423, −4.29346187142165139488742246961, −3.12956515425235147698166374435, −1.89119222994569974560395482668, 2.51249417092269402148520060559, 2.95934460272471343959043492971, 4.31628766683492577381286045067, 5.25183860905217522965912915675, 6.00566240183621525975990419921, 6.94700541341000120128559726841, 8.420629885558719381100149572501, 9.302921055704894233660319008726, 10.31960348077942384766911681420, 11.43796608301926771989756648090

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