Properties

Label 2-483-23.12-c1-0-5
Degree $2$
Conductor $483$
Sign $-0.920 - 0.391i$
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  + (−1.61 + 1.03i)2-s + (−0.142 + 0.989i)3-s + (0.698 − 1.53i)4-s + (2.13 − 0.625i)5-s + (−0.797 − 1.74i)6-s + (−0.654 − 0.755i)7-s + (−0.0867 − 0.603i)8-s + (−0.959 − 0.281i)9-s + (−2.79 + 3.22i)10-s + (−1.39 − 0.894i)11-s + (1.41 + 0.909i)12-s + (−2.99 + 3.45i)13-s + (1.84 + 0.540i)14-s + (0.316 + 2.19i)15-s + (2.96 + 3.42i)16-s + (2.73 + 5.99i)17-s + ⋯
L(s)  = 1  + (−1.14 + 0.733i)2-s + (−0.0821 + 0.571i)3-s + (0.349 − 0.765i)4-s + (0.952 − 0.279i)5-s + (−0.325 − 0.712i)6-s + (−0.247 − 0.285i)7-s + (−0.0306 − 0.213i)8-s + (−0.319 − 0.0939i)9-s + (−0.882 + 1.01i)10-s + (−0.419 − 0.269i)11-s + (0.408 + 0.262i)12-s + (−0.829 + 0.957i)13-s + (0.492 + 0.144i)14-s + (0.0815 + 0.567i)15-s + (0.742 + 0.856i)16-s + (0.663 + 1.45i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.920 - 0.391i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.920 - 0.391i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.920 - 0.391i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.121871 + 0.597192i\)
\(L(\frac12)\) \(\approx\) \(0.121871 + 0.597192i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.142 - 0.989i)T \)
7 \( 1 + (0.654 + 0.755i)T \)
23 \( 1 + (0.317 + 4.78i)T \)
good2 \( 1 + (1.61 - 1.03i)T + (0.830 - 1.81i)T^{2} \)
5 \( 1 + (-2.13 + 0.625i)T + (4.20 - 2.70i)T^{2} \)
11 \( 1 + (1.39 + 0.894i)T + (4.56 + 10.0i)T^{2} \)
13 \( 1 + (2.99 - 3.45i)T + (-1.85 - 12.8i)T^{2} \)
17 \( 1 + (-2.73 - 5.99i)T + (-11.1 + 12.8i)T^{2} \)
19 \( 1 + (2.49 - 5.46i)T + (-12.4 - 14.3i)T^{2} \)
29 \( 1 + (-2.61 - 5.73i)T + (-18.9 + 21.9i)T^{2} \)
31 \( 1 + (-1.43 - 9.99i)T + (-29.7 + 8.73i)T^{2} \)
37 \( 1 + (1.08 + 0.317i)T + (31.1 + 20.0i)T^{2} \)
41 \( 1 + (-2.60 + 0.765i)T + (34.4 - 22.1i)T^{2} \)
43 \( 1 + (0.992 - 6.90i)T + (-41.2 - 12.1i)T^{2} \)
47 \( 1 + 8.43T + 47T^{2} \)
53 \( 1 + (1.03 + 1.19i)T + (-7.54 + 52.4i)T^{2} \)
59 \( 1 + (-6.51 + 7.51i)T + (-8.39 - 58.3i)T^{2} \)
61 \( 1 + (1.71 + 11.9i)T + (-58.5 + 17.1i)T^{2} \)
67 \( 1 + (-6.90 + 4.43i)T + (27.8 - 60.9i)T^{2} \)
71 \( 1 + (0.127 - 0.0818i)T + (29.4 - 64.5i)T^{2} \)
73 \( 1 + (5.86 - 12.8i)T + (-47.8 - 55.1i)T^{2} \)
79 \( 1 + (-6.85 + 7.91i)T + (-11.2 - 78.1i)T^{2} \)
83 \( 1 + (-0.211 - 0.0620i)T + (69.8 + 44.8i)T^{2} \)
89 \( 1 + (1.01 - 7.05i)T + (-85.3 - 25.0i)T^{2} \)
97 \( 1 + (-14.9 + 4.39i)T + (81.6 - 52.4i)T^{2} \)
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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

−10.76123886864704389667903977552, −10.13467649988303873460886085850, −9.648715474652928407086712135082, −8.650055780318913145706788299418, −8.062863439206047700114188288525, −6.72551013965700699460930379992, −6.10849213847020339011788517881, −4.93710337461523322534217756981, −3.56387998694524732865771693946, −1.65580322879155336827063784753, 0.53410971837288042647816471638, 2.27822014543492410008933920239, 2.75913449224590318084996416213, 5.10493542910084715114264940989, 5.93167682151983863798845829206, 7.25971714673135440636934925606, 7.932873162807703505136858279140, 9.124385070671389994294924023065, 9.813077366527455459066320356722, 10.27318097635970325669364376630

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