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Properties

Label 1-473-473.16-r0-0-0
Degree $1$
Conductor $473$
Sign $-0.914 + 0.404i$
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  + (0.473 − 0.880i)2-s + (0.134 − 0.990i)3-s + (−0.550 − 0.834i)4-s + (0.753 − 0.657i)5-s + (−0.809 − 0.587i)6-s + (0.309 − 0.951i)7-s + (−0.995 + 0.0896i)8-s + (−0.963 − 0.266i)9-s + (−0.222 − 0.974i)10-s + (−0.900 + 0.433i)12-s + (−0.393 − 0.919i)13-s + (−0.691 − 0.722i)14-s + (−0.550 − 0.834i)15-s + (−0.393 + 0.919i)16-s + (−0.393 + 0.919i)17-s + (−0.691 + 0.722i)18-s + ⋯
L(s)  = 1  + (0.473 − 0.880i)2-s + (0.134 − 0.990i)3-s + (−0.550 − 0.834i)4-s + (0.753 − 0.657i)5-s + (−0.809 − 0.587i)6-s + (0.309 − 0.951i)7-s + (−0.995 + 0.0896i)8-s + (−0.963 − 0.266i)9-s + (−0.222 − 0.974i)10-s + (−0.900 + 0.433i)12-s + (−0.393 − 0.919i)13-s + (−0.691 − 0.722i)14-s + (−0.550 − 0.834i)15-s + (−0.393 + 0.919i)16-s + (−0.393 + 0.919i)17-s + (−0.691 + 0.722i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.914 + 0.404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(473\)    =    \(11 \cdot 43\)
Sign: $-0.914 + 0.404i$
Analytic conductor: \(2.19660\)
Root analytic conductor: \(2.19660\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{473} (16, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 473,\ (0:\ ),\ -0.914 + 0.404i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.3682319151 - 1.743039553i\)
\(L(\frac12)\) \(\approx\) \(-0.3682319151 - 1.743039553i\)
\(L(1)\) \(\approx\) \(0.6337896573 - 1.270512570i\)
\(L(1)\) \(\approx\) \(0.6337896573 - 1.270512570i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.473 - 0.880i)T \)
3 \( 1 + (0.134 - 0.990i)T \)
5 \( 1 + (0.753 - 0.657i)T \)
7 \( 1 + (0.309 - 0.951i)T \)
13 \( 1 + (-0.393 - 0.919i)T \)
17 \( 1 + (-0.393 + 0.919i)T \)
19 \( 1 + (0.936 + 0.351i)T \)
23 \( 1 + (0.623 + 0.781i)T \)
29 \( 1 + (0.134 + 0.990i)T \)
31 \( 1 + (0.473 - 0.880i)T \)
37 \( 1 + (0.309 - 0.951i)T \)
41 \( 1 + (-0.691 - 0.722i)T \)
47 \( 1 + (0.936 + 0.351i)T \)
53 \( 1 + (0.753 + 0.657i)T \)
59 \( 1 + (-0.995 - 0.0896i)T \)
61 \( 1 + (0.473 + 0.880i)T \)
67 \( 1 + (0.623 - 0.781i)T \)
71 \( 1 + (-0.963 + 0.266i)T \)
73 \( 1 + (0.858 + 0.512i)T \)
79 \( 1 + (-0.809 + 0.587i)T \)
83 \( 1 + (0.473 + 0.880i)T \)
89 \( 1 + (-0.900 - 0.433i)T \)
97 \( 1 + (-0.963 - 0.266i)T \)
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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

−24.71588615101630710401778412911, −23.3005655723249821988797068656, −22.381474266601808792108964546471, −21.935239383248321355493807331453, −21.28194927699233985732144228502, −20.49758623118383877248396365324, −18.939742205593542852686003224987, −18.12142077208098156390997216675, −17.28723554753385570509357048701, −16.42005629774203058619297165620, −15.53353863959485119017511500827, −14.89391404888891193276953577821, −14.1159655793940476457017744607, −13.50311370157347050083833939737, −11.98738091262101275105649289082, −11.314036967219663317276651091356, −9.88165424782119188766764666475, −9.23190286853241520442403782965, −8.429943751971813510036107666487, −7.07380883570655147867395946827, −6.18823400980704984071661022408, −5.17354510191491518333866779157, −4.60392148970529408413346600982, −3.1195311239932894375356413774, −2.442605977090265950815071163498, 0.86989036387821368322541418946, 1.609277922765735150216865162208, 2.73715285484321512963790176388, 3.921220445252788995153369322083, 5.21282289132343148170580179174, 5.87718067419485301864167608611, 7.162770856160692653625907220419, 8.24670560152036702935538753766, 9.26103071603202611819614133592, 10.26397216015670214730053647149, 11.119073400752968381880480799226, 12.2624246217005791012937232970, 12.891518260077590580115789788835, 13.610900592266190419528671932, 14.18128335084640157556189612717, 15.23960086278695187969302035466, 16.891893354917186244602390252751, 17.56211693520996927548258181066, 18.20737900967760905026580279593, 19.36663526423365861232530826680, 20.12253415219732893524885116088, 20.52953601441793362000003937529, 21.57086998477090802847133758200, 22.53631613014683737880001010419, 23.38243289205683852427169806743

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