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Properties

Label 2-483-161.19-c1-0-25
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: $\chi_{483} (19, \cdot )$
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

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

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