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Properties

Label 1-473-473.97-r0-0-0
Degree $1$
Conductor $473$
Sign $-0.0737 + 0.997i$
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  + (0.753 − 0.657i)2-s + (−0.995 − 0.0896i)3-s + (0.134 − 0.990i)4-s + (−0.0448 + 0.998i)5-s + (−0.809 + 0.587i)6-s + (0.309 + 0.951i)7-s + (−0.550 − 0.834i)8-s + (0.983 + 0.178i)9-s + (0.623 + 0.781i)10-s + (−0.222 + 0.974i)12-s + (−0.963 + 0.266i)13-s + (0.858 + 0.512i)14-s + (0.134 − 0.990i)15-s + (−0.963 − 0.266i)16-s + (−0.963 − 0.266i)17-s + (0.858 − 0.512i)18-s + ⋯
L(s)  = 1  + (0.753 − 0.657i)2-s + (−0.995 − 0.0896i)3-s + (0.134 − 0.990i)4-s + (−0.0448 + 0.998i)5-s + (−0.809 + 0.587i)6-s + (0.309 + 0.951i)7-s + (−0.550 − 0.834i)8-s + (0.983 + 0.178i)9-s + (0.623 + 0.781i)10-s + (−0.222 + 0.974i)12-s + (−0.963 + 0.266i)13-s + (0.858 + 0.512i)14-s + (0.134 − 0.990i)15-s + (−0.963 − 0.266i)16-s + (−0.963 − 0.266i)17-s + (0.858 − 0.512i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0737 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0737 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(473\)    =    \(11 \cdot 43\)
Sign: $-0.0737 + 0.997i$
Analytic conductor: \(2.19660\)
Root analytic conductor: \(2.19660\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{473} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 473,\ (0:\ ),\ -0.0737 + 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4560552405 + 0.4910065170i\)
\(L(\frac12)\) \(\approx\) \(0.4560552405 + 0.4910065170i\)
\(L(1)\) \(\approx\) \(0.8905032299 - 0.04173303939i\)
\(L(1)\) \(\approx\) \(0.8905032299 - 0.04173303939i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.753 - 0.657i)T \)
3 \( 1 + (-0.995 - 0.0896i)T \)
5 \( 1 + (-0.0448 + 0.998i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (-0.963 + 0.266i)T \)
17 \( 1 + (-0.963 - 0.266i)T \)
19 \( 1 + (-0.691 + 0.722i)T \)
23 \( 1 + (-0.900 + 0.433i)T \)
29 \( 1 + (-0.995 + 0.0896i)T \)
31 \( 1 + (0.753 - 0.657i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (0.858 + 0.512i)T \)
47 \( 1 + (-0.691 + 0.722i)T \)
53 \( 1 + (-0.0448 - 0.998i)T \)
59 \( 1 + (-0.550 + 0.834i)T \)
61 \( 1 + (0.753 + 0.657i)T \)
67 \( 1 + (-0.900 - 0.433i)T \)
71 \( 1 + (0.983 - 0.178i)T \)
73 \( 1 + (0.936 + 0.351i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (0.753 + 0.657i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + (0.983 + 0.178i)T \)
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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

−23.66449494513867201629905063884, −22.94392579275045601171832658816, −22.02432547981619696148655246579, −21.34852838454166005550912256461, −20.41284421340084189212334413602, −19.62443755354546244799876520242, −17.863574928091286792662253539075, −17.33830674077060823917402472987, −16.75935213305691604959993495296, −15.95849600642686186040415051732, −15.142688315133759747126699430747, −13.98066055009709834054276040225, −13.00164470104866316071081434746, −12.51036112093956284719622496457, −11.51327593829433935629152198381, −10.63841052672435496949639507191, −9.37398007184458445921166481886, −8.16055265037334178876703327525, −7.26003861765553512754737377507, −6.37184212115378994638462861052, −5.29772412259701914156051453022, −4.54327211686240415457230202803, −3.99582371918486668549728712727, −2.05464724620622233629715393871, −0.3021293583862608306721809784, 1.83216710667786949126480478389, 2.567938118341809285203750984718, 4.011190813860002210675692351, 4.92599920331662313747988107693, 5.97756914934199102692478531187, 6.52334317877656089859532033664, 7.729016794974123705713844609746, 9.47204301988332062776745815086, 10.2083088814146328451312842045, 11.24911791156219515168700018689, 11.67039582577075611412805548388, 12.499762113378453208553532026560, 13.478708828742147235556849387939, 14.64501295287729272200813896789, 15.178445183562859370147115022294, 16.10559214780954583575526546663, 17.44845768270187836728910674937, 18.27843307352232791375269881847, 18.878237176338234165487317829898, 19.69229577790754107626278874041, 21.088817272746083084904502713979, 21.70609485812159604936146003155, 22.43760779535459490273965462587, 22.76898532116376385421428740710, 24.057049221771241003865063926412

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