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Properties

Label 2-483-161.17-c1-0-8
Degree $2$
Conductor $483$
Sign $-0.399 - 0.916i$
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  + (−0.587 + 0.0561i)2-s + (0.690 + 0.723i)3-s + (−1.62 + 0.312i)4-s + (2.80 + 1.44i)5-s + (−0.446 − 0.386i)6-s + (−0.926 + 2.47i)7-s + (2.06 − 0.607i)8-s + (−0.0475 + 0.998i)9-s + (−1.72 − 0.692i)10-s + (−0.0160 + 0.168i)11-s + (−1.34 − 0.957i)12-s + (−3.12 + 0.449i)13-s + (0.405 − 1.50i)14-s + (0.888 + 3.02i)15-s + (1.88 − 0.754i)16-s + (0.814 + 2.35i)17-s + ⋯
L(s)  = 1  + (−0.415 + 0.0396i)2-s + (0.398 + 0.417i)3-s + (−0.810 + 0.156i)4-s + (1.25 + 0.646i)5-s + (−0.182 − 0.157i)6-s + (−0.350 + 0.936i)7-s + (0.731 − 0.214i)8-s + (−0.0158 + 0.332i)9-s + (−0.546 − 0.218i)10-s + (−0.00484 + 0.0507i)11-s + (−0.388 − 0.276i)12-s + (−0.866 + 0.124i)13-s + (0.108 − 0.403i)14-s + (0.229 + 0.781i)15-s + (0.470 − 0.188i)16-s + (0.197 + 0.570i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.399 - 0.916i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.399 - 0.916i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (178, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.399 - 0.916i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.623763 + 0.952613i\)
\(L(\frac12)\) \(\approx\) \(0.623763 + 0.952613i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.926 - 2.47i)T \)
23 \( 1 + (3.10 + 3.65i)T \)
good2 \( 1 + (0.587 - 0.0561i)T + (1.96 - 0.378i)T^{2} \)
5 \( 1 + (-2.80 - 1.44i)T + (2.90 + 4.07i)T^{2} \)
11 \( 1 + (0.0160 - 0.168i)T + (-10.8 - 2.08i)T^{2} \)
13 \( 1 + (3.12 - 0.449i)T + (12.4 - 3.66i)T^{2} \)
17 \( 1 + (-0.814 - 2.35i)T + (-13.3 + 10.5i)T^{2} \)
19 \( 1 + (-0.288 + 0.833i)T + (-14.9 - 11.7i)T^{2} \)
29 \( 1 + (4.98 - 5.74i)T + (-4.12 - 28.7i)T^{2} \)
31 \( 1 + (-5.54 - 1.34i)T + (27.5 + 14.2i)T^{2} \)
37 \( 1 + (-2.43 - 0.116i)T + (36.8 + 3.51i)T^{2} \)
41 \( 1 + (-1.83 - 2.85i)T + (-17.0 + 37.2i)T^{2} \)
43 \( 1 + (1.82 - 6.21i)T + (-36.1 - 23.2i)T^{2} \)
47 \( 1 + (-3.43 + 1.98i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (6.29 - 7.99i)T + (-12.4 - 51.5i)T^{2} \)
59 \( 1 + (-3.04 + 7.61i)T + (-42.7 - 40.7i)T^{2} \)
61 \( 1 + (-7.46 - 7.11i)T + (2.90 + 60.9i)T^{2} \)
67 \( 1 + (-11.3 + 8.08i)T + (21.9 - 63.3i)T^{2} \)
71 \( 1 + (-1.23 + 2.69i)T + (-46.4 - 53.6i)T^{2} \)
73 \( 1 + (2.56 + 13.3i)T + (-67.7 + 27.1i)T^{2} \)
79 \( 1 + (-2.92 - 3.72i)T + (-18.6 + 76.7i)T^{2} \)
83 \( 1 + (0.880 + 0.565i)T + (34.4 + 75.4i)T^{2} \)
89 \( 1 + (-3.73 - 15.3i)T + (-79.1 + 40.7i)T^{2} \)
97 \( 1 + (-2.00 + 1.28i)T + (40.2 - 88.2i)T^{2} \)
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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

−10.88286017207444233338920859971, −9.950582707160679490691212416912, −9.615329122574550135103522139391, −8.819331600423200013679425480292, −7.902250451321241609393019150270, −6.61109145865989359723791486767, −5.62339297724715487388469269057, −4.63770035674565637163773657934, −3.16854256423301359030573589543, −2.07573021266664962288932250271, 0.78551783868950822682004308481, 2.14056098694011280406612674900, 3.87151240396711804705428308197, 5.04148031705542096077803226521, 5.95282687333923353622623469451, 7.26119222115201721915509178856, 8.064837720447984116690132684140, 9.114907049935006354760870057493, 9.899457047923393352511003037918, 10.03227982400001070275785530730

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