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Properties

Label 1-475-475.61-r0-0-0
Degree $1$
Conductor $475$
Sign $0.357 - 0.933i$
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  + (0.848 − 0.529i)2-s + (0.961 + 0.275i)3-s + (0.438 − 0.898i)4-s + (0.961 − 0.275i)6-s + (−0.5 − 0.866i)7-s + (−0.104 − 0.994i)8-s + (0.848 + 0.529i)9-s + (0.669 + 0.743i)11-s + (0.669 − 0.743i)12-s + (0.0348 − 0.999i)13-s + (−0.882 − 0.469i)14-s + (−0.615 − 0.788i)16-s + (−0.997 − 0.0697i)17-s + 18-s + (−0.241 − 0.970i)21-s + (0.961 + 0.275i)22-s + ⋯
L(s)  = 1  + (0.848 − 0.529i)2-s + (0.961 + 0.275i)3-s + (0.438 − 0.898i)4-s + (0.961 − 0.275i)6-s + (−0.5 − 0.866i)7-s + (−0.104 − 0.994i)8-s + (0.848 + 0.529i)9-s + (0.669 + 0.743i)11-s + (0.669 − 0.743i)12-s + (0.0348 − 0.999i)13-s + (−0.882 − 0.469i)14-s + (−0.615 − 0.788i)16-s + (−0.997 − 0.0697i)17-s + 18-s + (−0.241 − 0.970i)21-s + (0.961 + 0.275i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.357 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.357 - 0.933i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.357 - 0.933i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ 0.357 - 0.933i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.440916661 - 1.678630370i\)
\(L(\frac12)\) \(\approx\) \(2.440916661 - 1.678630370i\)
\(L(1)\) \(\approx\) \(2.012794544 - 0.8220932517i\)
\(L(1)\) \(\approx\) \(2.012794544 - 0.8220932517i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.848 - 0.529i)T \)
3 \( 1 + (0.961 + 0.275i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (0.0348 - 0.999i)T \)
17 \( 1 + (-0.997 - 0.0697i)T \)
23 \( 1 + (0.990 + 0.139i)T \)
29 \( 1 + (-0.997 + 0.0697i)T \)
31 \( 1 + (0.913 + 0.406i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (-0.615 - 0.788i)T \)
43 \( 1 + (0.173 - 0.984i)T \)
47 \( 1 + (-0.997 + 0.0697i)T \)
53 \( 1 + (0.438 - 0.898i)T \)
59 \( 1 + (-0.374 + 0.927i)T \)
61 \( 1 + (0.990 + 0.139i)T \)
67 \( 1 + (-0.241 + 0.970i)T \)
71 \( 1 + (-0.719 + 0.694i)T \)
73 \( 1 + (0.0348 + 0.999i)T \)
79 \( 1 + (0.961 + 0.275i)T \)
83 \( 1 + (0.913 + 0.406i)T \)
89 \( 1 + (-0.615 + 0.788i)T \)
97 \( 1 + (-0.241 - 0.970i)T \)
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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

−24.332954833103654665772439971180, −23.21047837970756414537090558776, −22.17480933833370501387041301192, −21.54823049132332836720068491358, −20.827563574847060499202000966530, −19.68992628136976563581740791329, −19.06233113171665512713325643617, −18.070535742723296965005554847825, −16.81623033083461315644910235919, −16.026950637582007723750602704851, −15.12870737053529536527459551826, −14.539944942061311672395091126636, −13.54334446092084566900848772123, −13.009000795083032460343032041156, −11.98077577878409356811794504497, −11.172949501148410943757024382943, −9.320228236841779203787887253962, −8.88304117200744967433468931284, −7.86694031634942811751977884434, −6.66408668584205970777487523503, −6.226779190547045329309869126824, −4.75300877616106726080288653508, −3.74076397175122694352588219443, −2.8521253057834026545425319578, −1.87970888996269409304019537757, 1.24362181575014741701542818849, 2.4795978941025160127758859210, 3.45044166136530856661072096244, 4.19323252367596976880022449163, 5.135274897613408757540892667420, 6.6689182279973893295665240041, 7.2947569033222726879120335447, 8.72281857871599488037769497294, 9.785578286381308547984263458768, 10.341262394808906380581583802698, 11.36221775388344969194588952311, 12.67622109553924980710226815418, 13.25342933840421598571848643297, 13.96377468616443377289609528904, 15.02315465719487414051779925000, 15.41366732172396367474306142697, 16.56553945486880545177441995477, 17.73809097683539692442790657551, 19.065523702050487505672832933784, 19.673041849514352545136611172579, 20.41610875251406347196415155279, 20.792201261979101996111831710937, 22.15280388931867316069267177991, 22.53460750729050623647366459915, 23.5209742541065136365629969108

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