Properties

Label 1-475-475.336-r1-0-0
Degree $1$
Conductor $475$
Sign $-0.729 - 0.684i$
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  + (0.882 + 0.469i)2-s + (0.241 + 0.970i)3-s + (0.559 + 0.829i)4-s + (−0.241 + 0.970i)6-s + (−0.5 − 0.866i)7-s + (0.104 + 0.994i)8-s + (−0.882 + 0.469i)9-s + (0.669 + 0.743i)11-s + (−0.669 + 0.743i)12-s + (−0.848 − 0.529i)13-s + (−0.0348 − 0.999i)14-s + (−0.374 + 0.927i)16-s + (0.438 + 0.898i)17-s − 18-s + (0.719 − 0.694i)21-s + (0.241 + 0.970i)22-s + ⋯
L(s)  = 1  + (0.882 + 0.469i)2-s + (0.241 + 0.970i)3-s + (0.559 + 0.829i)4-s + (−0.241 + 0.970i)6-s + (−0.5 − 0.866i)7-s + (0.104 + 0.994i)8-s + (−0.882 + 0.469i)9-s + (0.669 + 0.743i)11-s + (−0.669 + 0.743i)12-s + (−0.848 − 0.529i)13-s + (−0.0348 − 0.999i)14-s + (−0.374 + 0.927i)16-s + (0.438 + 0.898i)17-s − 18-s + (0.719 − 0.694i)21-s + (0.241 + 0.970i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.729 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.729 - 0.684i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (336, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ -0.729 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.6434248405 + 1.626656428i\)
\(L(\frac12)\) \(\approx\) \(-0.6434248405 + 1.626656428i\)
\(L(1)\) \(\approx\) \(1.073187219 + 1.021598475i\)
\(L(1)\) \(\approx\) \(1.073187219 + 1.021598475i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.882 + 0.469i)T \)
3 \( 1 + (0.241 + 0.970i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.669 + 0.743i)T \)
13 \( 1 + (-0.848 - 0.529i)T \)
17 \( 1 + (0.438 + 0.898i)T \)
23 \( 1 + (-0.615 + 0.788i)T \)
29 \( 1 + (-0.438 + 0.898i)T \)
31 \( 1 + (-0.913 - 0.406i)T \)
37 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (0.374 - 0.927i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (0.438 - 0.898i)T \)
53 \( 1 + (-0.559 - 0.829i)T \)
59 \( 1 + (-0.990 + 0.139i)T \)
61 \( 1 + (-0.615 + 0.788i)T \)
67 \( 1 + (0.719 + 0.694i)T \)
71 \( 1 + (-0.961 - 0.275i)T \)
73 \( 1 + (0.848 - 0.529i)T \)
79 \( 1 + (0.241 + 0.970i)T \)
83 \( 1 + (0.913 + 0.406i)T \)
89 \( 1 + (0.374 + 0.927i)T \)
97 \( 1 + (0.719 - 0.694i)T \)
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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

−23.01772718812408595552416378598, −22.22238648154428913281717771318, −21.60310810487367531582739194813, −20.41673737450978382521675618635, −19.714527645196348745380586079171, −18.82120169222021080258137703586, −18.53032153638553797315094232183, −16.993183821507115179824586808609, −16.07993512669354765862248158667, −14.91937930313748770648728121257, −14.21893318367240311373181671559, −13.51618771527581700724001034164, −12.44922234438628270625251406575, −11.99697071753441320597017414110, −11.22892806936501253921645434121, −9.73154141283279006192959339176, −8.98890723196922497640481820399, −7.65454771150113843600364475530, −6.54346810406898943147419982261, −5.99426228010994125132686885979, −4.853675088395949568904501350171, −3.41156363926995397934685166077, −2.63472777119316475757117163481, −1.66037698660148782561616206299, −0.284658118861487184050681019204, 2.02079500212646199587699619356, 3.46008235691925315966721039492, 3.90093596801487170998784365747, 4.96548859799099128177970144530, 5.87378032957299557167098187203, 7.10374633990292719411139720806, 7.84009472233428589804071970900, 9.141689574470721688631801449822, 10.09474811347683687549341041363, 10.91678006630013106336800233741, 12.10414988421387360929295790363, 12.92389703114437223821327495397, 13.990147514877055253798788737966, 14.69493102601074924924846794860, 15.31875091747583093186556925317, 16.36275999345702698538820997444, 16.94615858655205675935654491094, 17.66243036580614090419325581810, 19.64653774411482255819278396528, 19.96443449356211430727995587050, 20.853597109817745292261212476755, 21.862038956043460279213332573118, 22.36059350072924241957321866617, 23.14455812301540168249883281257, 23.96203254803801596800917825606

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