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Properties

Label 1-473-473.312-r0-0-0
Degree $1$
Conductor $473$
Sign $-0.998 - 0.0517i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.995 + 0.0896i)2-s + (−0.393 + 0.919i)3-s + (0.983 − 0.178i)4-s + (−0.550 − 0.834i)5-s + (0.309 − 0.951i)6-s + (−0.809 + 0.587i)7-s + (−0.963 + 0.266i)8-s + (−0.691 − 0.722i)9-s + (0.623 + 0.781i)10-s + (−0.222 + 0.974i)12-s + (0.936 + 0.351i)13-s + (0.753 − 0.657i)14-s + (0.983 − 0.178i)15-s + (0.936 − 0.351i)16-s + (0.936 − 0.351i)17-s + (0.753 + 0.657i)18-s + ⋯
L(s)  = 1  + (−0.995 + 0.0896i)2-s + (−0.393 + 0.919i)3-s + (0.983 − 0.178i)4-s + (−0.550 − 0.834i)5-s + (0.309 − 0.951i)6-s + (−0.809 + 0.587i)7-s + (−0.963 + 0.266i)8-s + (−0.691 − 0.722i)9-s + (0.623 + 0.781i)10-s + (−0.222 + 0.974i)12-s + (0.936 + 0.351i)13-s + (0.753 − 0.657i)14-s + (0.983 − 0.178i)15-s + (0.936 − 0.351i)16-s + (0.936 − 0.351i)17-s + (0.753 + 0.657i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0517i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0517i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(473\)    =    \(11 \cdot 43\)
Sign: $-0.998 - 0.0517i$
Analytic conductor: \(2.19660\)
Root analytic conductor: \(2.19660\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{473} (312, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 473,\ (0:\ ),\ -0.998 - 0.0517i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.004775989778 + 0.1843928970i\)
\(L(\frac12)\) \(\approx\) \(0.004775989778 + 0.1843928970i\)
\(L(1)\) \(\approx\) \(0.4144560511 + 0.1412434678i\)
\(L(1)\) \(\approx\) \(0.4144560511 + 0.1412434678i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.995 + 0.0896i)T \)
3 \( 1 + (-0.393 + 0.919i)T \)
5 \( 1 + (-0.550 - 0.834i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (0.936 + 0.351i)T \)
17 \( 1 + (0.936 - 0.351i)T \)
19 \( 1 + (0.473 + 0.880i)T \)
23 \( 1 + (-0.900 + 0.433i)T \)
29 \( 1 + (-0.393 - 0.919i)T \)
31 \( 1 + (-0.995 + 0.0896i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.753 - 0.657i)T \)
47 \( 1 + (0.473 + 0.880i)T \)
53 \( 1 + (-0.550 + 0.834i)T \)
59 \( 1 + (-0.963 - 0.266i)T \)
61 \( 1 + (-0.995 - 0.0896i)T \)
67 \( 1 + (-0.900 - 0.433i)T \)
71 \( 1 + (-0.691 + 0.722i)T \)
73 \( 1 + (-0.0448 + 0.998i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (-0.995 - 0.0896i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + (-0.691 - 0.722i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.51925894360014568294803909970, −22.71934667329399960363682294761, −21.831134113730169780323013555850, −20.364956580778427985685325913271, −19.72924312121033415421235344113, −19.02928971997589873660249985405, −18.28311738852062698023544785512, −17.73389118919013232183156011637, −16.50884763117522541950648429426, −16.0762617825064854801983127678, −14.85680975323617235510040470027, −13.75470817767454969038238837333, −12.68625418766423601403728687238, −11.89005429196596115345541740701, −10.85156491415880793088532426134, −10.47361870376315643643803483153, −9.13586555095711340906663070934, −7.95747615738653154507340242262, −7.33870340535755631219613901416, −6.56366349784150611382604911668, −5.77085734555933997158314772031, −3.64049850766160003511478270385, −2.84909353635350945272846170753, −1.45631324138053885783448011804, −0.16064571715399636793483475658, 1.38979390298568248359197114805, 3.13394498234122482714953919078, 3.99607076108347116419071384973, 5.57586728555146801536132795061, 6.01705461826602719332664364908, 7.51519027696837857573766130103, 8.51555547426724466501422122586, 9.31759007508575215662970213650, 9.87200429996662684194229745237, 11.00620475161317620235770277796, 11.90680543924337087132544069614, 12.435229181799202891837427458377, 14.078848018772112776844820387498, 15.38110660896502714470201395944, 15.90224444382229608616961593305, 16.43458072918851592217091658730, 17.15094589477509611085529073810, 18.362752695888528972032104859885, 19.04955571308618964716131232702, 20.09755106433472392740312374384, 20.73384039216158352444521605892, 21.43441991191353476320120237560, 22.64632402373857732697790887300, 23.42561610898612696165826781856, 24.35830363034620007253766820893

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