Properties

Label 1-475-475.71-r1-0-0
Degree $1$
Conductor $475$
Sign $-0.986 - 0.161i$
Analytic cond. $51.0458$
Root an. cond. $51.0458$
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  + (0.997 − 0.0697i)2-s + (−0.0348 + 0.999i)3-s + (0.990 − 0.139i)4-s + (0.0348 + 0.999i)6-s + (−0.5 + 0.866i)7-s + (0.978 − 0.207i)8-s + (−0.997 − 0.0697i)9-s + (−0.104 + 0.994i)11-s + (0.104 + 0.994i)12-s + (−0.559 + 0.829i)13-s + (−0.438 + 0.898i)14-s + (0.961 − 0.275i)16-s + (−0.374 − 0.927i)17-s − 18-s + (−0.848 − 0.529i)21-s + (−0.0348 + 0.999i)22-s + ⋯
L(s)  = 1  + (0.997 − 0.0697i)2-s + (−0.0348 + 0.999i)3-s + (0.990 − 0.139i)4-s + (0.0348 + 0.999i)6-s + (−0.5 + 0.866i)7-s + (0.978 − 0.207i)8-s + (−0.997 − 0.0697i)9-s + (−0.104 + 0.994i)11-s + (0.104 + 0.994i)12-s + (−0.559 + 0.829i)13-s + (−0.438 + 0.898i)14-s + (0.961 − 0.275i)16-s + (−0.374 − 0.927i)17-s − 18-s + (−0.848 − 0.529i)21-s + (−0.0348 + 0.999i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.986 - 0.161i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.986 - 0.161i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (71, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ -0.986 - 0.161i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1479921715 + 1.819739178i\)
\(L(\frac12)\) \(\approx\) \(-0.1479921715 + 1.819739178i\)
\(L(1)\) \(\approx\) \(1.297737941 + 0.7682925679i\)
\(L(1)\) \(\approx\) \(1.297737941 + 0.7682925679i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.997 - 0.0697i)T \)
3 \( 1 + (-0.0348 + 0.999i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (-0.559 + 0.829i)T \)
17 \( 1 + (-0.374 - 0.927i)T \)
23 \( 1 + (-0.719 + 0.694i)T \)
29 \( 1 + (0.374 - 0.927i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (-0.961 + 0.275i)T \)
43 \( 1 + (0.173 + 0.984i)T \)
47 \( 1 + (-0.374 + 0.927i)T \)
53 \( 1 + (-0.990 + 0.139i)T \)
59 \( 1 + (0.241 - 0.970i)T \)
61 \( 1 + (-0.719 + 0.694i)T \)
67 \( 1 + (-0.848 + 0.529i)T \)
71 \( 1 + (0.882 - 0.469i)T \)
73 \( 1 + (0.559 + 0.829i)T \)
79 \( 1 + (-0.0348 + 0.999i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + (-0.961 - 0.275i)T \)
97 \( 1 + (-0.848 - 0.529i)T \)
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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

−23.40282762020115737751914594679, −22.336381833178713075142641752046, −21.83024784652447676048821273070, −20.40703177228513485181057533125, −19.88886150508931247243367991424, −19.17772513715715568148164196097, −17.99673994824286564937891318908, −16.9350653200634874056583535959, −16.38924979864113647752712065101, −15.157785226603300228825247667570, −14.19623072354161466535684797777, −13.541827495907268087098604725265, −12.79612955658955504924138938084, −12.15109033307003590719013243380, −10.94734935670146248906798087008, −10.3522678263534814045084475697, −8.527394159743831479517984282240, −7.6954963996198630569995926107, −6.747792618527329707680451858697, −6.08145435500998061871262088843, −5.051126999434553792051695908723, −3.65856164389992914623850959440, −2.8847869951366938582442589172, −1.60164657233034520064212729237, −0.2946196104563016178283331039, 2.09816918565726066009463743217, 2.872161566258256590039612788107, 4.10736771640477859311751013198, 4.81058846052155453483095108129, 5.74312324027728615326418893248, 6.68313553163968430543775181743, 7.88846507117064825514517081923, 9.5255853689515118479991767550, 9.7120721475317452924387343441, 11.184300511177054307583543140501, 11.80537862653296694817432694092, 12.64879680287372441258312943384, 13.78850245135500716555560681738, 14.66798554297795863391337628443, 15.386756938631037885121373413672, 16.00369844905555415172691104494, 16.822405042082485625960969430836, 17.999109672846647042326257392665, 19.319174663837424641558520153457, 20.05118337505755703806796231606, 20.896457776897290553440258784036, 21.66181998210849021876491463756, 22.303631752776592515606249088599, 22.899399005713987637597462892698, 23.86442308502783077430907925622

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