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Properties

Label 1-473-473.147-r1-0-0
Degree $1$
Conductor $473$
Sign $0.554 - 0.832i$
Analytic cond. $50.8309$
Root an. cond. $50.8309$
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  + (0.550 − 0.834i)2-s + (−0.251 + 0.967i)3-s + (−0.393 − 0.919i)4-s + (0.925 + 0.379i)5-s + (0.669 + 0.743i)6-s + (−0.913 − 0.406i)7-s + (−0.983 − 0.178i)8-s + (−0.873 − 0.486i)9-s + (0.826 − 0.563i)10-s + (0.988 − 0.149i)12-s + (−0.280 + 0.959i)13-s + (−0.842 + 0.538i)14-s + (−0.599 + 0.800i)15-s + (−0.691 + 0.722i)16-s + (0.971 + 0.237i)17-s + (−0.887 + 0.460i)18-s + ⋯
L(s)  = 1  + (0.550 − 0.834i)2-s + (−0.251 + 0.967i)3-s + (−0.393 − 0.919i)4-s + (0.925 + 0.379i)5-s + (0.669 + 0.743i)6-s + (−0.913 − 0.406i)7-s + (−0.983 − 0.178i)8-s + (−0.873 − 0.486i)9-s + (0.826 − 0.563i)10-s + (0.988 − 0.149i)12-s + (−0.280 + 0.959i)13-s + (−0.842 + 0.538i)14-s + (−0.599 + 0.800i)15-s + (−0.691 + 0.722i)16-s + (0.971 + 0.237i)17-s + (−0.887 + 0.460i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 473 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.554 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(473\)    =    \(11 \cdot 43\)
Sign: $0.554 - 0.832i$
Analytic conductor: \(50.8309\)
Root analytic conductor: \(50.8309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{473} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 473,\ (1:\ ),\ 0.554 - 0.832i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.945945509 - 1.041456523i\)
\(L(\frac12)\) \(\approx\) \(1.945945509 - 1.041456523i\)
\(L(1)\) \(\approx\) \(1.256069323 - 0.3132431436i\)
\(L(1)\) \(\approx\) \(1.256069323 - 0.3132431436i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
43 \( 1 \)
good2 \( 1 + (0.550 - 0.834i)T \)
3 \( 1 + (-0.251 + 0.967i)T \)
5 \( 1 + (0.925 + 0.379i)T \)
7 \( 1 + (-0.913 - 0.406i)T \)
13 \( 1 + (-0.280 + 0.959i)T \)
17 \( 1 + (0.971 + 0.237i)T \)
19 \( 1 + (-0.193 - 0.981i)T \)
23 \( 1 + (0.955 + 0.294i)T \)
29 \( 1 + (-0.251 - 0.967i)T \)
31 \( 1 + (-0.447 - 0.894i)T \)
37 \( 1 + (0.104 + 0.994i)T \)
41 \( 1 + (-0.0448 - 0.998i)T \)
47 \( 1 + (0.753 - 0.657i)T \)
53 \( 1 + (0.791 + 0.611i)T \)
59 \( 1 + (0.983 - 0.178i)T \)
61 \( 1 + (0.447 - 0.894i)T \)
67 \( 1 + (-0.733 - 0.680i)T \)
71 \( 1 + (-0.0149 + 0.999i)T \)
73 \( 1 + (0.999 - 0.0299i)T \)
79 \( 1 + (0.669 - 0.743i)T \)
83 \( 1 + (0.998 - 0.0598i)T \)
89 \( 1 + (-0.365 - 0.930i)T \)
97 \( 1 + (0.858 - 0.512i)T \)
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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

−23.79418299076590559355779309075, −22.90925034396677221118453734023, −22.4186170427231016491869309672, −21.4381435973987684256420137188, −20.50476072076249434904499833766, −19.323648758890940238333469477, −18.29039369151065064491760724288, −17.772474388163027221591525981236, −16.62543048098058230867676067759, −16.40148727870537782505375324967, −14.8867266322544475116979523330, −14.187745934770393474935482653, −13.14405620146847628701616928815, −12.69026455831986951845612166038, −12.11166961966932607863782375491, −10.4946144905180161090048891527, −9.31089499390065778783365015253, −8.43561649533726924727002573734, −7.3924653333465405093545386748, −6.5212735243705355066546782691, −5.6368574235102582905786550440, −5.22821002277094949575930618495, −3.37214423608545762464331475673, −2.470459479296451302206874746886, −0.89381510514602679620802218504, 0.62846238429998511166300548197, 2.245634037539754053969031321440, 3.20901586983706718171362953147, 4.09279457648331473449323641160, 5.1778980154993044193049815832, 6.02318888894620666494588629910, 6.91286253454428447245667188331, 9.03976197860659314235219470243, 9.58993743886258178651947678419, 10.26041305737863986076905988043, 11.06495158184217277229676179203, 11.97961937492088726806954905518, 13.15943104843100037014057303987, 13.81821981481425626441116434580, 14.73474298866634651559915187773, 15.488541802558575094280893069172, 16.77818804146531166903127828557, 17.28436136478277085798296502127, 18.66117105088426973342654802039, 19.30717358312611703800295220901, 20.38434723294140347425704851672, 21.12147247565699806806687474413, 21.78689933916950206164563743449, 22.38164303803138658926553562740, 23.14660338970019517405493034669

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