↑

Properties

Label 2-483-161.10-c1-0-4
Degree $2$
Conductor $483$
Sign $-0.834 - 0.550i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0478 − 0.138i)2-s + (0.458 + 0.888i)3-s + (1.55 + 1.22i)4-s + (−2.41 + 0.231i)5-s + (0.144 − 0.0208i)6-s + (−2.60 + 0.466i)7-s + (0.490 − 0.314i)8-s + (−0.580 + 0.814i)9-s + (−0.0838 + 0.345i)10-s + (−5.38 + 1.86i)11-s + (−0.374 + 1.94i)12-s + (−0.253 − 0.863i)13-s + (−0.0602 + 0.382i)14-s + (−1.31 − 2.04i)15-s + (0.912 + 3.76i)16-s + (0.714 + 0.286i)17-s + ⋯
L(s)  = 1  + (0.0338 − 0.0978i)2-s + (0.264 + 0.513i)3-s + (0.777 + 0.611i)4-s + (−1.08 + 0.103i)5-s + (0.0591 − 0.00850i)6-s + (−0.984 + 0.176i)7-s + (0.173 − 0.111i)8-s + (−0.193 + 0.271i)9-s + (−0.0265 + 0.109i)10-s + (−1.62 + 0.561i)11-s + (−0.108 + 0.560i)12-s + (−0.0703 − 0.239i)13-s + (−0.0160 + 0.102i)14-s + (−0.339 − 0.527i)15-s + (0.228 + 0.940i)16-s + (0.173 + 0.0694i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.834 - 0.550i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.834 - 0.550i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.834 - 0.550i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.834 - 0.550i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.248450 + 0.827337i\)
\(L(\frac12)\) \(\approx\) \(0.248450 + 0.827337i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.458 - 0.888i)T \)
7 \( 1 + (2.60 - 0.466i)T \)
23 \( 1 + (1.38 - 4.59i)T \)
good2 \( 1 + (-0.0478 + 0.138i)T + (-1.57 - 1.23i)T^{2} \)
5 \( 1 + (2.41 - 0.231i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (5.38 - 1.86i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (0.253 + 0.863i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-0.714 - 0.286i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-1.13 + 0.452i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (0.344 + 2.39i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (-10.1 + 0.482i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-2.10 - 1.49i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (6.60 + 3.01i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (5.42 - 8.44i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-0.365 + 0.210i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-8.49 - 8.90i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (-3.92 - 0.953i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (3.27 + 1.68i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-1.80 - 9.38i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (0.668 + 0.771i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (4.24 - 5.39i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (2.46 - 2.58i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-0.321 - 0.704i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.0252 - 0.530i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-6.03 + 13.2i)T + (-63.5 - 73.3i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.47295757938499870113616180074, −10.35215911074585117347727204496, −9.857405123699286631477682527579, −8.380355905229269552139111167064, −7.77868503587390088487285387006, −7.02330493078773466057759252050, −5.73095544949173119950519354218, −4.37958401470785129159025229063, −3.30073390813644063601848427163, −2.61546745865564099372273743145, 0.46133703218872077315684135662, 2.48349948385440291233847287817, 3.43493902182437148037711291239, 4.99749390416905244056374502380, 6.14458798332068992470021892281, 6.97995078943952605776425783723, 7.80473411053514089027807622784, 8.542764404253892412951321616976, 10.01010190013945881928639096738, 10.53058772632296356279982471054

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