Properties

Label 2-483-161.2-c1-0-14
Degree $2$
Conductor $483$
Sign $0.882 - 0.469i$
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  + (1.42 − 0.733i)2-s + (−0.786 + 0.618i)3-s + (0.325 − 0.457i)4-s + (−0.994 + 0.948i)5-s + (−0.664 + 1.45i)6-s + (1.69 − 2.03i)7-s + (−0.327 + 2.27i)8-s + (0.235 − 0.971i)9-s + (−0.719 + 2.07i)10-s + (3.16 + 1.62i)11-s + (0.0267 + 0.561i)12-s + (3.84 + 4.43i)13-s + (0.916 − 4.13i)14-s + (0.195 − 1.36i)15-s + (1.57 + 4.54i)16-s + (−4.09 + 0.390i)17-s + ⋯
L(s)  = 1  + (1.00 − 0.518i)2-s + (−0.453 + 0.356i)3-s + (0.162 − 0.228i)4-s + (−0.444 + 0.424i)5-s + (−0.271 + 0.594i)6-s + (0.639 − 0.768i)7-s + (−0.115 + 0.805i)8-s + (0.0785 − 0.323i)9-s + (−0.227 + 0.657i)10-s + (0.953 + 0.491i)11-s + (0.00771 + 0.161i)12-s + (1.06 + 1.22i)13-s + (0.245 − 1.10i)14-s + (0.0504 − 0.351i)15-s + (0.393 + 1.13i)16-s + (−0.992 + 0.0948i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.469i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 - 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.882 - 0.469i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (163, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.882 - 0.469i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94343 + 0.484547i\)
\(L(\frac12)\) \(\approx\) \(1.94343 + 0.484547i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.786 - 0.618i)T \)
7 \( 1 + (-1.69 + 2.03i)T \)
23 \( 1 + (-4.78 - 0.303i)T \)
good2 \( 1 + (-1.42 + 0.733i)T + (1.16 - 1.62i)T^{2} \)
5 \( 1 + (0.994 - 0.948i)T + (0.237 - 4.99i)T^{2} \)
11 \( 1 + (-3.16 - 1.62i)T + (6.38 + 8.96i)T^{2} \)
13 \( 1 + (-3.84 - 4.43i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (4.09 - 0.390i)T + (16.6 - 3.21i)T^{2} \)
19 \( 1 + (0.804 + 0.0768i)T + (18.6 + 3.59i)T^{2} \)
29 \( 1 + (-1.00 + 2.19i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (-2.88 + 1.15i)T + (22.4 - 21.3i)T^{2} \)
37 \( 1 + (-2.02 + 8.34i)T + (-32.8 - 16.9i)T^{2} \)
41 \( 1 + (11.3 + 3.32i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (0.484 + 3.37i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + (4.44 - 7.69i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (1.11 + 0.214i)T + (49.2 + 19.6i)T^{2} \)
59 \( 1 + (-1.91 + 5.52i)T + (-46.3 - 36.4i)T^{2} \)
61 \( 1 + (-0.376 - 0.295i)T + (14.3 + 59.2i)T^{2} \)
67 \( 1 + (-0.516 + 10.8i)T + (-66.6 - 6.36i)T^{2} \)
71 \( 1 + (-9.21 - 5.91i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (-6.56 + 9.21i)T + (-23.8 - 68.9i)T^{2} \)
79 \( 1 + (14.1 - 2.73i)T + (73.3 - 29.3i)T^{2} \)
83 \( 1 + (8.83 - 2.59i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (-1.19 - 0.478i)T + (64.4 + 61.4i)T^{2} \)
97 \( 1 + (-13.5 - 3.96i)T + (81.6 + 52.4i)T^{2} \)
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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

−11.32946378371539986545920273947, −10.71789851573418904169560500290, −9.323433389305044726141529196339, −8.445348499108105101799854862025, −7.10861786538051762730973456520, −6.34814203986307879316556783535, −4.91570021161456888448694666084, −4.17725925577200544703717898369, −3.58935208721317269711877607451, −1.78330439088643136597412083982, 1.09951215393367014601020819886, 3.20630120331406265357319267159, 4.47600084426957177066800226531, 5.22108107577979884970955725565, 6.15837077604787025331317459445, 6.81155833752158433794013640272, 8.295912041724067625238190415553, 8.720356295346419132357155438538, 10.16709596518406416031557581684, 11.33463677345714107967225270132

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