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Properties

Label 1-475-475.258-r1-0-0
Degree $1$
Conductor $475$
Sign $0.938 - 0.345i$
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  + (0.406 − 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

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

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.938 - 0.345i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (258, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ 0.938 - 0.345i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7076036733 - 0.1260714958i\)
\(L(\frac12)\) \(\approx\) \(0.7076036733 - 0.1260714958i\)
\(L(1)\) \(\approx\) \(0.6259309142 - 0.6341999267i\)
\(L(1)\) \(\approx\) \(0.6259309142 - 0.6341999267i\)

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

−23.702707825452531681498984960451, −22.52701910577548599376777923581, −22.29086862143430562847165378750, −21.170747158716942754414716609068, −20.82644557393641452369341060860, −19.286472161528341718493910316056, −18.20507044922778285370055516028, −17.50455778332283043241884007165, −16.5458846263484540190042716504, −15.60946252350487671015769817743, −15.364578588892498637616593236058, −14.454848427460780912826482565508, −13.259764638167528401523472783698, −12.53670910413601022800744042182, −11.31377469173999538486558211831, −10.4726060125338711061096685922, −9.09176986785225317690731445630, −8.674918573947557627774488170129, −7.59512088540704990446135975228, −6.056970315519696939452021847092, −5.630562134987049827591396740914, −4.73045075347842622440942499705, −3.55196243097902084968856638430, −2.695514167531834272863102671567, −0.19565975307056820620523651362, 0.97203320368467471499851332555, 1.9817438945779492501109253762, 3.01569372284592407338449625050, 4.34350794974327856534349064886, 5.18424788300356589103207014683, 6.50218449521113705361429562165, 7.25249776787220723340483658157, 8.47271159758990937525638183535, 9.56239089515615012690167710446, 10.70068993167308149950639145886, 11.29926887420138841116480450125, 12.24204478329368868549394024899, 13.21022189997156977525997776802, 13.61528583973716956338254184458, 14.4754656158460631974546728042, 15.72557800105191273338264688103, 17.036182704552887730795246379601, 17.759554351230479967799869133568, 18.669998044977333420338575211577, 19.247109744509658282776698903025, 20.3446618432632281587025591991, 20.688377283034338109014191374530, 21.93090625562218534306982103630, 22.97597431826004900148887472827, 23.344559820418425540479646045637

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