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Properties

Label 1-473-473.207-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: $\chi_{473} (207, \cdot )$
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

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

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