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Properties

Label 1-473-473.471-r0-0-0
Degree $1$
Conductor $473$
Sign $0.973 + 0.229i$
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  + (−0.963 + 0.266i)2-s + (0.936 − 0.351i)3-s + (0.858 − 0.512i)4-s + (0.983 − 0.178i)5-s + (−0.809 + 0.587i)6-s + (0.309 + 0.951i)7-s + (−0.691 + 0.722i)8-s + (0.753 − 0.657i)9-s + (−0.900 + 0.433i)10-s + (0.623 − 0.781i)12-s + (0.473 + 0.880i)13-s + (−0.550 − 0.834i)14-s + (0.858 − 0.512i)15-s + (0.473 − 0.880i)16-s + (0.473 − 0.880i)17-s + (−0.550 + 0.834i)18-s + ⋯
L(s)  = 1  + (−0.963 + 0.266i)2-s + (0.936 − 0.351i)3-s + (0.858 − 0.512i)4-s + (0.983 − 0.178i)5-s + (−0.809 + 0.587i)6-s + (0.309 + 0.951i)7-s + (−0.691 + 0.722i)8-s + (0.753 − 0.657i)9-s + (−0.900 + 0.433i)10-s + (0.623 − 0.781i)12-s + (0.473 + 0.880i)13-s + (−0.550 − 0.834i)14-s + (0.858 − 0.512i)15-s + (0.473 − 0.880i)16-s + (0.473 − 0.880i)17-s + (−0.550 + 0.834i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.973 + 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(473\)    =    \(11 \cdot 43\)
Sign: $0.973 + 0.229i$
Analytic conductor: \(2.19660\)
Root analytic conductor: \(2.19660\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{473} (471, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 473,\ (0:\ ),\ 0.973 + 0.229i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.551466639 + 0.1803118784i\)
\(L(\frac12)\) \(\approx\) \(1.551466639 + 0.1803118784i\)
\(L(1)\) \(\approx\) \(1.192222325 + 0.07658061334i\)
\(L(1)\) \(\approx\) \(1.192222325 + 0.07658061334i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.963 + 0.266i)T \)
3 \( 1 + (0.936 - 0.351i)T \)
5 \( 1 + (0.983 - 0.178i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (0.473 + 0.880i)T \)
17 \( 1 + (0.473 - 0.880i)T \)
19 \( 1 + (-0.995 - 0.0896i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (0.936 + 0.351i)T \)
31 \( 1 + (-0.963 + 0.266i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (-0.550 - 0.834i)T \)
47 \( 1 + (-0.995 - 0.0896i)T \)
53 \( 1 + (0.983 + 0.178i)T \)
59 \( 1 + (-0.691 - 0.722i)T \)
61 \( 1 + (-0.963 - 0.266i)T \)
67 \( 1 + (-0.222 - 0.974i)T \)
71 \( 1 + (0.753 + 0.657i)T \)
73 \( 1 + (0.134 - 0.990i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.963 - 0.266i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
97 \( 1 + (0.753 - 0.657i)T \)
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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

−24.17683809184328238896050982037, −22.85966273485197838142228851365, −21.463931276119145245735182888890, −21.235251918094615730023287503643, −20.248499284356358483811331111041, −19.7362785221423267498113771226, −18.647304156774176544616599727596, −17.905368519814290671599846947868, −16.98726277752767476842813227852, −16.335978659108783254398405718022, −15.06910407946629875837943661771, −14.433751734305772426043067606148, −13.293511888025701376686598798, −12.660630598399644741793004668886, −10.89895955011595530468142893410, −10.407959027301097585245351224271, −9.8047268314191281367524924799, −8.629190128113474430885305171252, −8.06960036276463213799226503355, −6.99849077862268641416027313320, −5.944260935604061805924901037083, −4.30104735338394155560616335332, −3.25425212517944914177592987874, −2.21442936261200313553163803489, −1.251560188077597276434347680896, 1.469792759340273852455360106823, 2.07696086167567281467236039211, 3.11146657998215120666644634357, 4.96971434773759480404400552563, 6.11255032667564173653241200447, 6.88231940781404096652774234607, 8.04691397413709257846706621287, 8.92093069095732958824518289081, 9.310168649532152596721810532432, 10.28586987418225345234702405311, 11.55872108293926908342898236745, 12.4781606393277420139513734193, 13.69617659438803616421490152911, 14.39029133943358597407656741610, 15.260722629927649743464365569375, 16.14921222653152237630221136104, 17.162745917532064548524694736370, 18.204207292398202376609228061, 18.50394387791416973251695673900, 19.431973826178497139506393829893, 20.37535396250558017253198611028, 21.24729878775872726287686005358, 21.6194480936405835210444458863, 23.45272475173966438296097100880, 24.24678149166857976727664678034

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