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Properties

Label 2-483-161.4-c1-0-22
Degree $2$
Conductor $483$
Sign $0.393 + 0.919i$
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  + (−0.0919 + 0.129i)2-s + (0.235 − 0.971i)3-s + (0.645 + 1.86i)4-s + (0.0626 − 1.31i)5-s + (0.103 + 0.119i)6-s + (0.568 − 2.58i)7-s + (−0.604 − 0.177i)8-s + (−0.888 − 0.458i)9-s + (0.164 + 0.129i)10-s + (−1.86 − 2.62i)11-s + (1.96 − 0.187i)12-s + (0.717 − 4.99i)13-s + (0.281 + 0.311i)14-s + (−1.26 − 0.371i)15-s + (−3.02 + 2.37i)16-s + (−2.10 + 0.406i)17-s + ⋯
L(s)  = 1  + (−0.0650 + 0.0913i)2-s + (0.136 − 0.561i)3-s + (0.322 + 0.933i)4-s + (0.0280 − 0.588i)5-s + (0.0423 + 0.0489i)6-s + (0.214 − 0.976i)7-s + (−0.213 − 0.0627i)8-s + (−0.296 − 0.152i)9-s + (0.0519 + 0.0408i)10-s + (−0.562 − 0.790i)11-s + (0.567 − 0.0541i)12-s + (0.198 − 1.38i)13-s + (0.0752 + 0.0831i)14-s + (−0.326 − 0.0958i)15-s + (−0.756 + 0.594i)16-s + (−0.510 + 0.0984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.393 + 0.919i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.393 + 0.919i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.393 + 0.919i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.393 + 0.919i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.20729 - 0.796841i\)
\(L(\frac12)\) \(\approx\) \(1.20729 - 0.796841i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.235 + 0.971i)T \)
7 \( 1 + (-0.568 + 2.58i)T \)
23 \( 1 + (-3.12 - 3.64i)T \)
good2 \( 1 + (0.0919 - 0.129i)T + (-0.654 - 1.89i)T^{2} \)
5 \( 1 + (-0.0626 + 1.31i)T + (-4.97 - 0.475i)T^{2} \)
11 \( 1 + (1.86 + 2.62i)T + (-3.59 + 10.3i)T^{2} \)
13 \( 1 + (-0.717 + 4.99i)T + (-12.4 - 3.66i)T^{2} \)
17 \( 1 + (2.10 - 0.406i)T + (15.7 - 6.31i)T^{2} \)
19 \( 1 + (-5.62 - 1.08i)T + (17.6 + 7.06i)T^{2} \)
29 \( 1 + (6.22 + 7.18i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-2.58 + 2.46i)T + (1.47 - 30.9i)T^{2} \)
37 \( 1 + (-3.24 - 1.67i)T + (21.4 + 30.1i)T^{2} \)
41 \( 1 + (-7.80 - 5.01i)T + (17.0 + 37.2i)T^{2} \)
43 \( 1 + (6.63 - 1.94i)T + (36.1 - 23.2i)T^{2} \)
47 \( 1 + (0.750 + 1.30i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.34 - 1.73i)T + (38.3 + 36.5i)T^{2} \)
59 \( 1 + (7.83 + 6.15i)T + (13.9 + 57.3i)T^{2} \)
61 \( 1 + (-1.87 - 7.72i)T + (-54.2 + 27.9i)T^{2} \)
67 \( 1 + (-9.58 - 0.915i)T + (65.7 + 12.6i)T^{2} \)
71 \( 1 + (2.89 + 6.33i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-4.06 - 11.7i)T + (-57.3 + 45.1i)T^{2} \)
79 \( 1 + (11.4 - 4.58i)T + (57.1 - 54.5i)T^{2} \)
83 \( 1 + (-2.44 + 1.57i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (-8.39 - 8.00i)T + (4.23 + 88.8i)T^{2} \)
97 \( 1 + (-6.27 - 4.03i)T + (40.2 + 88.2i)T^{2} \)
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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

−11.11735398095095503074470723077, −9.892528979144501970209069731275, −8.740417496622771388710399623531, −7.74868745210560338989030352552, −7.66242877074761167389487388354, −6.28176119638627914479506230002, −5.16945015208204506767338587731, −3.75247526582914344988884926157, −2.80491424968519620474844419112, −0.919903289924043550507649867489, 1.93186110415345414328172549937, 2.93582755980121106848759057428, 4.65858183800809102984285094103, 5.37333040556174864338617565033, 6.53631062552939153236702318279, 7.30617296613452463421264718005, 8.921178041043657978184434055818, 9.307830496174062361874988233462, 10.31446323253668771875856563757, 11.08543931501128950120974702617

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