↑

Properties

Label 1-483-483.359-r0-0-0
Degree $1$
Conductor $483$
Sign $-0.370 + 0.928i$
Analytic cond. $2.24304$
Root an. cond. $2.24304$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.888 + 0.458i)2-s + (0.580 + 0.814i)4-s + (0.723 + 0.690i)5-s + (0.142 + 0.989i)8-s + (0.327 + 0.945i)10-s + (−0.888 + 0.458i)11-s + (−0.654 + 0.755i)13-s + (−0.327 + 0.945i)16-s + (−0.995 − 0.0950i)17-s + (0.995 − 0.0950i)19-s + (−0.142 + 0.989i)20-s − 22-s + (0.0475 + 0.998i)25-s + (−0.928 + 0.371i)26-s + (−0.415 − 0.909i)29-s + ⋯
L(s)  = 1  + (0.888 + 0.458i)2-s + (0.580 + 0.814i)4-s + (0.723 + 0.690i)5-s + (0.142 + 0.989i)8-s + (0.327 + 0.945i)10-s + (−0.888 + 0.458i)11-s + (−0.654 + 0.755i)13-s + (−0.327 + 0.945i)16-s + (−0.995 − 0.0950i)17-s + (0.995 − 0.0950i)19-s + (−0.142 + 0.989i)20-s − 22-s + (0.0475 + 0.998i)25-s + (−0.928 + 0.371i)26-s + (−0.415 − 0.909i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.370 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.370 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.370 + 0.928i$
Analytic conductor: \(2.24304\)
Root analytic conductor: \(2.24304\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (359, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 483,\ (0:\ ),\ -0.370 + 0.928i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.305585517 + 1.925601261i\)
\(L(\frac12)\) \(\approx\) \(1.305585517 + 1.925601261i\)
\(L(1)\) \(\approx\) \(1.507814833 + 0.9587695505i\)
\(L(1)\) \(\approx\) \(1.507814833 + 0.9587695505i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good2 \( 1 + (0.888 + 0.458i)T \)
5 \( 1 + (0.723 + 0.690i)T \)
11 \( 1 + (-0.888 + 0.458i)T \)
13 \( 1 + (-0.654 + 0.755i)T \)
17 \( 1 + (-0.995 - 0.0950i)T \)
19 \( 1 + (0.995 - 0.0950i)T \)
29 \( 1 + (-0.415 - 0.909i)T \)
31 \( 1 + (0.928 + 0.371i)T \)
37 \( 1 + (-0.235 - 0.971i)T \)
41 \( 1 + (0.959 - 0.281i)T \)
43 \( 1 + (0.142 - 0.989i)T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.981 - 0.189i)T \)
59 \( 1 + (0.327 + 0.945i)T \)
61 \( 1 + (0.786 - 0.618i)T \)
67 \( 1 + (-0.0475 - 0.998i)T \)
71 \( 1 + (-0.841 + 0.540i)T \)
73 \( 1 + (0.580 + 0.814i)T \)
79 \( 1 + (-0.981 - 0.189i)T \)
83 \( 1 + (-0.959 - 0.281i)T \)
89 \( 1 + (0.928 - 0.371i)T \)
97 \( 1 + (0.959 - 0.281i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.51708174678739301947365275613, −22.41995510333819736969050887148, −21.87861834599834298963151194570, −20.952748090117154449966872882410, −20.34412679352828905218193728106, −19.598574039388571303485775303, −18.41683712847969988206797505, −17.60440316100760669180694915344, −16.42474809363344037704001650979, −15.68777823279626062673001953758, −14.739211550466187528323478180889, −13.65988320318787227014908390135, −13.17126418525308090375847657061, −12.38365309767421676055366756981, −11.35844196755483227518444529524, −10.34759315697012377462653686888, −9.66758486478024697348935089961, −8.46666462323263449435679979751, −7.23173761787747804665471216012, −5.9910149116823838372506778402, −5.286322180881152837527171077342, −4.52014802563271174896571475663, −3.08892457239032704113174389488, −2.25557955565660227068126475897, −0.93425368499399522959342787435, 2.13176472684529840960757985196, 2.700763174571455861640219565361, 4.07435680785842247516344927841, 5.08921741161316364506799422308, 5.95112242424889510746963817130, 7.001428657785748228925007768383, 7.53843120880713379224693867245, 8.96813783766896633794721202476, 10.0405561893240489260214855890, 11.04204180836879377230497406895, 11.94015925977025688105682786840, 12.99504446128962562844740995563, 13.75230112172644119792618803529, 14.40825676865681284004093586571, 15.378599123071440211070063999434, 16.03004846088950351192162809443, 17.28138236343599528501773455917, 17.76633182707925510598405044715, 18.85008188028707528159685431254, 20.01970155964977696632452029461, 20.99112689472371380634182118003, 21.572983118174165865902273699479, 22.49386614041482407345503501415, 22.95534947981393350020221215769, 24.17238217464017678508647447993

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