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Properties

Label 1-475-475.144-r0-0-0
Degree $1$
Conductor $475$
Sign $0.345 + 0.938i$
Analytic cond. $2.20589$
Root an. cond. $2.20589$
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  + (−0.913 − 0.406i)2-s + (0.978 − 0.207i)3-s + (0.669 + 0.743i)4-s + (−0.978 − 0.207i)6-s − 7-s + (−0.309 − 0.951i)8-s + (0.913 − 0.406i)9-s + (−0.809 + 0.587i)11-s + (0.809 + 0.587i)12-s + (−0.913 + 0.406i)13-s + (0.913 + 0.406i)14-s + (−0.104 + 0.994i)16-s + (−0.669 + 0.743i)17-s − 18-s + (−0.978 + 0.207i)21-s + (0.978 − 0.207i)22-s + ⋯
L(s)  = 1  + (−0.913 − 0.406i)2-s + (0.978 − 0.207i)3-s + (0.669 + 0.743i)4-s + (−0.978 − 0.207i)6-s − 7-s + (−0.309 − 0.951i)8-s + (0.913 − 0.406i)9-s + (−0.809 + 0.587i)11-s + (0.809 + 0.587i)12-s + (−0.913 + 0.406i)13-s + (0.913 + 0.406i)14-s + (−0.104 + 0.994i)16-s + (−0.669 + 0.743i)17-s − 18-s + (−0.978 + 0.207i)21-s + (0.978 − 0.207i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.345 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.345 + 0.938i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (144, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ 0.345 + 0.938i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6031625096 + 0.4207375013i\)
\(L(\frac12)\) \(\approx\) \(0.6031625096 + 0.4207375013i\)
\(L(1)\) \(\approx\) \(0.7585175479 + 0.01704222984i\)
\(L(1)\) \(\approx\) \(0.7585175479 + 0.01704222984i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.913 - 0.406i)T \)
3 \( 1 + (0.978 - 0.207i)T \)
7 \( 1 - T \)
11 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.913 + 0.406i)T \)
17 \( 1 + (-0.669 + 0.743i)T \)
23 \( 1 + (0.104 + 0.994i)T \)
29 \( 1 + (0.669 + 0.743i)T \)
31 \( 1 + (0.309 + 0.951i)T \)
37 \( 1 + (0.809 + 0.587i)T \)
41 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.669 - 0.743i)T \)
53 \( 1 + (-0.669 - 0.743i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + (-0.104 - 0.994i)T \)
67 \( 1 + (0.978 + 0.207i)T \)
71 \( 1 + (-0.978 + 0.207i)T \)
73 \( 1 + (-0.913 - 0.406i)T \)
79 \( 1 + (-0.978 + 0.207i)T \)
83 \( 1 + (-0.309 - 0.951i)T \)
89 \( 1 + (-0.104 - 0.994i)T \)
97 \( 1 + (0.978 - 0.207i)T \)
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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

−24.07921043893550017382565099066, −22.84822252654272196269084796265, −21.88992109285785155189016587227, −20.704228578565649801802827794732, −20.161253964266693416316889844646, −19.178459068169756650544437893332, −18.82858991853604024642087597223, −17.73981535385541773286121424679, −16.6194263139201326581778882397, −15.851554958956416178374861456132, −15.33192691361926216485509395816, −14.27555270780734938177823383553, −13.40010983756249362651742068830, −12.398448028149312684919709353317, −10.953076696524988880306350420376, −10.07914551636843927324872809192, −9.45833000745359721921547520960, −8.56388846109412654655893638069, −7.69526223573284007567297866440, −6.88310356800632895312471737819, −5.75484178680124661269493299037, −4.4711474542189304744831870942, −2.84976264684154785139971955522, −2.42011098147039045405243452052, −0.460682184441655416987611149669, 1.524400054376718521966213690801, 2.58577285428176918498341369145, 3.28968574954954062181392578444, 4.55884763156441932870685273909, 6.455334251787536712071345145443, 7.1919393957573356966027394843, 8.05680201259130477702933571111, 9.018172552358810779882005419607, 9.80109992739491812375983183874, 10.364499482290011421929107671602, 11.77478505317252371143060233154, 12.8082218436539881795996354118, 13.16787191973235251715151769663, 14.605065159695543830260691370416, 15.54429683179586783223265702559, 16.17726543343400800181786429292, 17.36463369916924038147387397889, 18.16733883791919720917026595543, 19.073326767248594593759389172537, 19.74273108379778915090016827926, 20.138400677432746515399082162816, 21.39994154044558682501563288496, 21.80121235169613269218167347427, 23.24435042191332138405360107640, 24.24972815608014363674241036989

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