Properties

Label 2-483-161.66-c1-0-15
Degree $2$
Conductor $483$
Sign $0.951 + 0.308i$
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  + (1.29 − 1.02i)2-s + (−0.814 + 0.580i)3-s + (0.171 − 0.707i)4-s + (0.682 + 0.131i)5-s + (−0.465 + 1.58i)6-s + (2.26 − 1.36i)7-s + (0.872 + 1.91i)8-s + (0.327 − 0.945i)9-s + (1.02 − 0.526i)10-s + (−2.09 + 2.66i)11-s + (0.270 + 0.676i)12-s + (0.864 − 1.34i)13-s + (1.54 − 4.08i)14-s + (−0.632 + 0.288i)15-s + (4.37 + 2.25i)16-s + (3.99 − 3.81i)17-s + ⋯
L(s)  = 1  + (0.918 − 0.721i)2-s + (−0.470 + 0.334i)3-s + (0.0858 − 0.353i)4-s + (0.305 + 0.0588i)5-s + (−0.189 + 0.647i)6-s + (0.856 − 0.516i)7-s + (0.308 + 0.675i)8-s + (0.109 − 0.315i)9-s + (0.322 − 0.166i)10-s + (−0.631 + 0.803i)11-s + (0.0781 + 0.195i)12-s + (0.239 − 0.373i)13-s + (0.412 − 1.09i)14-s + (−0.163 + 0.0745i)15-s + (1.09 + 0.564i)16-s + (0.970 − 0.924i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.951 + 0.308i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.951 + 0.308i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (388, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.951 + 0.308i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.21200 - 0.349221i\)
\(L(\frac12)\) \(\approx\) \(2.21200 - 0.349221i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.814 - 0.580i)T \)
7 \( 1 + (-2.26 + 1.36i)T \)
23 \( 1 + (-3.31 - 3.46i)T \)
good2 \( 1 + (-1.29 + 1.02i)T + (0.471 - 1.94i)T^{2} \)
5 \( 1 + (-0.682 - 0.131i)T + (4.64 + 1.85i)T^{2} \)
11 \( 1 + (2.09 - 2.66i)T + (-2.59 - 10.6i)T^{2} \)
13 \( 1 + (-0.864 + 1.34i)T + (-5.40 - 11.8i)T^{2} \)
17 \( 1 + (-3.99 + 3.81i)T + (0.808 - 16.9i)T^{2} \)
19 \( 1 + (-4.05 - 3.86i)T + (0.904 + 18.9i)T^{2} \)
29 \( 1 + (2.35 + 0.692i)T + (24.3 + 15.6i)T^{2} \)
31 \( 1 + (-0.692 + 7.25i)T + (-30.4 - 5.86i)T^{2} \)
37 \( 1 + (8.00 + 2.77i)T + (29.0 + 22.8i)T^{2} \)
41 \( 1 + (-8.73 - 7.57i)T + (5.83 + 40.5i)T^{2} \)
43 \( 1 + (6.54 + 2.98i)T + (28.1 + 32.4i)T^{2} \)
47 \( 1 + (1.10 - 0.640i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (9.75 - 0.464i)T + (52.7 - 5.03i)T^{2} \)
59 \( 1 + (3.95 + 7.66i)T + (-34.2 + 48.0i)T^{2} \)
61 \( 1 + (2.25 - 3.16i)T + (-19.9 - 57.6i)T^{2} \)
67 \( 1 + (0.828 - 2.06i)T + (-48.4 - 46.2i)T^{2} \)
71 \( 1 + (-2.03 - 14.1i)T + (-68.1 + 20.0i)T^{2} \)
73 \( 1 + (15.6 + 3.78i)T + (64.8 + 33.4i)T^{2} \)
79 \( 1 + (-11.8 - 0.564i)T + (78.6 + 7.50i)T^{2} \)
83 \( 1 + (6.10 + 7.04i)T + (-11.8 + 82.1i)T^{2} \)
89 \( 1 + (11.2 - 1.07i)T + (87.3 - 16.8i)T^{2} \)
97 \( 1 + (3.89 - 4.49i)T + (-13.8 - 96.0i)T^{2} \)
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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

−11.21710554953141501226620575982, −10.23934254220578162369002780648, −9.619636643352851275403955357557, −7.962556570447102091638270778784, −7.44625898273574688712860899621, −5.68453701382076941156932060765, −5.13702818735938417452050892083, −4.17501252178837286854374836530, −3.09081091809230621640114795811, −1.61953315662085363168073550310, 1.42767310337510951344058938334, 3.27123789072306345351210028753, 4.77430500128353028774035698978, 5.43068221380571623615755087991, 6.08564957569615423892981123485, 7.15220495692069933049517935548, 8.053211538627721748515784470559, 9.089147558168081617201367352593, 10.37013120042693304773874364978, 11.10571284749390816324059152220

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