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Properties

Label 1-473-473.64-r0-0-0
Degree $1$
Conductor $473$
Sign $-0.438 - 0.898i$
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  + (−0.0448 − 0.998i)2-s + (−0.550 − 0.834i)3-s + (−0.995 + 0.0896i)4-s + (0.473 − 0.880i)5-s + (−0.809 + 0.587i)6-s + (0.309 + 0.951i)7-s + (0.134 + 0.990i)8-s + (−0.393 + 0.919i)9-s + (−0.900 − 0.433i)10-s + (0.623 + 0.781i)12-s + (0.983 + 0.178i)13-s + (0.936 − 0.351i)14-s + (−0.995 + 0.0896i)15-s + (0.983 − 0.178i)16-s + (0.983 − 0.178i)17-s + (0.936 + 0.351i)18-s + ⋯
L(s)  = 1  + (−0.0448 − 0.998i)2-s + (−0.550 − 0.834i)3-s + (−0.995 + 0.0896i)4-s + (0.473 − 0.880i)5-s + (−0.809 + 0.587i)6-s + (0.309 + 0.951i)7-s + (0.134 + 0.990i)8-s + (−0.393 + 0.919i)9-s + (−0.900 − 0.433i)10-s + (0.623 + 0.781i)12-s + (0.983 + 0.178i)13-s + (0.936 − 0.351i)14-s + (−0.995 + 0.0896i)15-s + (0.983 − 0.178i)16-s + (0.983 − 0.178i)17-s + (0.936 + 0.351i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.438 - 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(473\)    =    \(11 \cdot 43\)
Sign: $-0.438 - 0.898i$
Analytic conductor: \(2.19660\)
Root analytic conductor: \(2.19660\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{473} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 473,\ (0:\ ),\ -0.438 - 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6195804887 - 0.9918194385i\)
\(L(\frac12)\) \(\approx\) \(0.6195804887 - 0.9918194385i\)
\(L(1)\) \(\approx\) \(0.7182080366 - 0.6443437339i\)
\(L(1)\) \(\approx\) \(0.7182080366 - 0.6443437339i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
43 \( 1 \)
good2 \( 1 + (-0.0448 - 0.998i)T \)
3 \( 1 + (-0.550 - 0.834i)T \)
5 \( 1 + (0.473 - 0.880i)T \)
7 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (0.983 + 0.178i)T \)
17 \( 1 + (0.983 - 0.178i)T \)
19 \( 1 + (0.858 + 0.512i)T \)
23 \( 1 + (-0.222 - 0.974i)T \)
29 \( 1 + (-0.550 + 0.834i)T \)
31 \( 1 + (-0.0448 - 0.998i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (0.936 - 0.351i)T \)
47 \( 1 + (0.858 + 0.512i)T \)
53 \( 1 + (0.473 + 0.880i)T \)
59 \( 1 + (0.134 - 0.990i)T \)
61 \( 1 + (-0.0448 + 0.998i)T \)
67 \( 1 + (-0.222 + 0.974i)T \)
71 \( 1 + (-0.393 - 0.919i)T \)
73 \( 1 + (-0.691 - 0.722i)T \)
79 \( 1 + (-0.809 - 0.587i)T \)
83 \( 1 + (-0.0448 + 0.998i)T \)
89 \( 1 + (0.623 - 0.781i)T \)
97 \( 1 + (-0.393 + 0.919i)T \)
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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

−23.88677439353373512111377022842, −23.086578272492097132838660323676, −22.786533231144300119221704561283, −21.64302384739389868777371771587, −21.111472603951348771436037573735, −19.85085354713555454059289305962, −18.57472138619216482955659077674, −17.775301989281201755811022183517, −17.28262352159557252689825489441, −16.29624341185304109487191485119, −15.624332771543471110577878390908, −14.63906970581656545211336113352, −14.0193074183981577350260961687, −13.17082936883455247900617007311, −11.602526034246899958801281785, −10.6784759030480475661524677758, −9.984741192423585325401074856773, −9.16626537535259174275379822939, −7.81117901329248563087215113971, −6.99527096068022097547686247380, −5.94466606560638128198134001928, −5.33856990944135270953043243582, −4.02486478537083884713875159256, −3.36121421604124642886117336296, −1.07138321405516579942785788315, 1.00717482799751055857380996173, 1.75860893751069749149518715659, 2.85287013263882700704341142948, 4.39843572147107484284341040013, 5.51353496966966604688259189899, 5.92881746767706336286506015992, 7.74475629944939380638022353426, 8.55249517776322652769217386044, 9.36172140537388258041192984405, 10.49772462413461395601476443141, 11.56671963395211448197835472742, 12.15931892299500434454658587450, 12.83479388744834383453466531475, 13.67078979694321013819752879614, 14.49688705841968852030203695025, 16.167730272433412957417755608482, 16.88112997838431722850800180140, 17.93018911103479088401172645230, 18.467223664701037728662042635742, 19.06826665822090071410320412469, 20.414544184679722233253700426108, 20.80177368612320613511571333242, 21.88300438783802594544086142650, 22.53027662995934608844629816325, 23.554032408941814227479498090516

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