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Properties

Label 1-475-475.217-r0-0-0
Degree $1$
Conductor $475$
Sign $0.998 + 0.0559i$
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  + (0.406 − 0.913i)2-s + (−0.207 − 0.978i)3-s + (−0.669 − 0.743i)4-s + (−0.978 − 0.207i)6-s + i·7-s + (−0.951 + 0.309i)8-s + (−0.913 + 0.406i)9-s + (−0.809 + 0.587i)11-s + (−0.587 + 0.809i)12-s + (0.406 + 0.913i)13-s + (0.913 + 0.406i)14-s + (−0.104 + 0.994i)16-s + (0.743 + 0.669i)17-s + i·18-s + (0.978 − 0.207i)21-s + (0.207 + 0.978i)22-s + ⋯
L(s)  = 1  + (0.406 − 0.913i)2-s + (−0.207 − 0.978i)3-s + (−0.669 − 0.743i)4-s + (−0.978 − 0.207i)6-s + i·7-s + (−0.951 + 0.309i)8-s + (−0.913 + 0.406i)9-s + (−0.809 + 0.587i)11-s + (−0.587 + 0.809i)12-s + (0.406 + 0.913i)13-s + (0.913 + 0.406i)14-s + (−0.104 + 0.994i)16-s + (0.743 + 0.669i)17-s + i·18-s + (0.978 − 0.207i)21-s + (0.207 + 0.978i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0559i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.998 + 0.0559i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ 0.998 + 0.0559i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8459539881 + 0.02369477031i\)
\(L(\frac12)\) \(\approx\) \(0.8459539881 + 0.02369477031i\)
\(L(1)\) \(\approx\) \(0.8211853877 - 0.4029521875i\)
\(L(1)\) \(\approx\) \(0.8211853877 - 0.4029521875i\)

Euler product

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

−23.41971697436377981195014714328, −23.21019885884682521758507701211, −22.29304299264346076499397307166, −21.3210325453563225693369050657, −20.758014985077021160097450049691, −19.78873132825599102878726308546, −18.2722586255792557256044568090, −17.61325517425603172501690088855, −16.59922529842799786835081712460, −16.09422191932315321636633668270, −15.39127315040417869704667333785, −14.278459796934326923931597507990, −13.739432803949279291238051323054, −12.684294649126265272012845975799, −11.50499811716623639960622947435, −10.42839725367134363835701193918, −9.76830293019715723416125457933, −8.422528684133330224619125240328, −7.82540097493395581374272459828, −6.55903902064313406735438573239, −5.54278126965694231407998467254, −4.86011031950453012552870725522, −3.72007577274207532277880154592, −3.07345243825916415671961520633, −0.41844049356610621301489362251, 1.531680420238164946499484793182, 2.21868492474309744430750112559, 3.298317475449627746152932997387, 4.74351227228201426929056769242, 5.693991757521318549588422905644, 6.44842042507846618948421192001, 7.890664873428203863263819664603, 8.75246780062833625078232823152, 9.83463851993787603112456469335, 10.89674338948868162840900597393, 11.873325236361845718337481564299, 12.35522796803287133263042501708, 13.15791421084538379725289225668, 14.06657443286588995542266025328, 14.89035351953942521438404108438, 15.99689307844725070461303288312, 17.33464841679291824852233924991, 18.28484275463681092616076003187, 18.68248948303506560156743737849, 19.4586179287823718566773125292, 20.44196388020053611462068362730, 21.32249814566235444928797755431, 22.09433774276113000470659462358, 22.99222133200621648632107161079, 23.791797572960139417640330297212

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