↑

Properties

Label 1-473-473.4-r0-0-0
Degree $1$
Conductor $473$
Sign $-0.313 - 0.949i$
Analytic cond. $2.19660$
Root an. cond. $2.19660$
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  + (0.858 − 0.512i)2-s + (0.753 − 0.657i)3-s + (0.473 − 0.880i)4-s + (0.936 − 0.351i)5-s + (0.309 − 0.951i)6-s + (−0.809 + 0.587i)7-s + (−0.0448 − 0.998i)8-s + (0.134 − 0.990i)9-s + (0.623 − 0.781i)10-s + (−0.222 − 0.974i)12-s + (−0.550 + 0.834i)13-s + (−0.393 + 0.919i)14-s + (0.473 − 0.880i)15-s + (−0.550 − 0.834i)16-s + (−0.550 − 0.834i)17-s + (−0.393 − 0.919i)18-s + ⋯
L(s)  = 1  + (0.858 − 0.512i)2-s + (0.753 − 0.657i)3-s + (0.473 − 0.880i)4-s + (0.936 − 0.351i)5-s + (0.309 − 0.951i)6-s + (−0.809 + 0.587i)7-s + (−0.0448 − 0.998i)8-s + (0.134 − 0.990i)9-s + (0.623 − 0.781i)10-s + (−0.222 − 0.974i)12-s + (−0.550 + 0.834i)13-s + (−0.393 + 0.919i)14-s + (0.473 − 0.880i)15-s + (−0.550 − 0.834i)16-s + (−0.550 − 0.834i)17-s + (−0.393 − 0.919i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.313 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(473\)    =    \(11 \cdot 43\)
Sign: $-0.313 - 0.949i$
Analytic conductor: \(2.19660\)
Root analytic conductor: \(2.19660\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{473} (4, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 473,\ (0:\ ),\ -0.313 - 0.949i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.741324197 - 2.407450858i\)
\(L(\frac12)\) \(\approx\) \(1.741324197 - 2.407450858i\)
\(L(1)\) \(\approx\) \(1.773652750 - 1.272829622i\)
\(L(1)\) \(\approx\) \(1.773652750 - 1.272829622i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.858 - 0.512i)T \)
3 \( 1 + (0.753 - 0.657i)T \)
5 \( 1 + (0.936 - 0.351i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.550 + 0.834i)T \)
17 \( 1 + (-0.550 - 0.834i)T \)
19 \( 1 + (0.983 + 0.178i)T \)
23 \( 1 + (-0.900 - 0.433i)T \)
29 \( 1 + (0.753 + 0.657i)T \)
31 \( 1 + (0.858 - 0.512i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (-0.393 + 0.919i)T \)
47 \( 1 + (0.983 + 0.178i)T \)
53 \( 1 + (0.936 + 0.351i)T \)
59 \( 1 + (-0.0448 + 0.998i)T \)
61 \( 1 + (0.858 + 0.512i)T \)
67 \( 1 + (-0.900 + 0.433i)T \)
71 \( 1 + (0.134 + 0.990i)T \)
73 \( 1 + (-0.963 - 0.266i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.858 + 0.512i)T \)
89 \( 1 + (-0.222 + 0.974i)T \)
97 \( 1 + (0.134 - 0.990i)T \)
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

−24.2740624544140139065224121851, −22.99490961635519343084623738707, −22.22214962900330226574105503073, −21.821269508606812286968878504421, −20.82298256351587136228810097036, −20.08597291200862650436181605321, −19.30448000294268863596948204143, −17.70694200253833400387628459640, −17.13922583621244562863578520362, −15.995658297829896971017959006124, −15.46686473093265463877828256327, −14.454770567014616512146101120813, −13.71276949648558459578826632630, −13.26257609823684815569063154125, −12.144899595924556913960196042375, −10.603822827901171611799965968288, −10.09153464305001606888406436622, −9.00672459592179779576612897169, −7.867578722326185966974396712375, −6.9521555248455384387937650986, −5.92762358714511640171034440310, −5.00333071844843553061661708398, −3.82914588195386195870109926469, −3.055564753933208504215019976814, −2.12228860338864071788744299962, 1.20898536226087282162637479817, 2.38065208526136743889721862369, 2.85643172486628735148619537943, 4.25241611069508338637573242825, 5.4270481913595224356224279209, 6.39969414552703329532088038346, 7.05592813245212075706138616932, 8.70027659012891077749159096166, 9.53267812426908269577417182251, 10.11029368227599402165471506477, 11.88327099227938927040020728638, 12.216485611742539324958905693861, 13.3491758772283482025487644545, 13.75684662661517456717589986239, 14.54430653339550402670918017107, 15.63009168009969161165753779743, 16.48108967736818506300753766832, 17.96276601071448669742711395170, 18.62038968972873218112683622389, 19.515514287252804996752760842982, 20.25699150906251640696429803815, 20.97655363996080357592130411502, 21.930686904166857743960666039056, 22.47549914324195187229278745529, 23.70594211926987960986261703156

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