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Properties

Label 1-475-475.21-r1-0-0
Degree $1$
Conductor $475$
Sign $-0.856 - 0.516i$
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  + (−0.559 − 0.829i)2-s + (0.882 − 0.469i)3-s + (−0.374 + 0.927i)4-s + (−0.882 − 0.469i)6-s + (−0.5 + 0.866i)7-s + (0.978 − 0.207i)8-s + (0.559 − 0.829i)9-s + (−0.104 + 0.994i)11-s + (0.104 + 0.994i)12-s + (−0.438 − 0.898i)13-s + (0.997 − 0.0697i)14-s + (−0.719 − 0.694i)16-s + (−0.615 + 0.788i)17-s − 18-s + (−0.0348 + 0.999i)21-s + (0.882 − 0.469i)22-s + ⋯
L(s)  = 1  + (−0.559 − 0.829i)2-s + (0.882 − 0.469i)3-s + (−0.374 + 0.927i)4-s + (−0.882 − 0.469i)6-s + (−0.5 + 0.866i)7-s + (0.978 − 0.207i)8-s + (0.559 − 0.829i)9-s + (−0.104 + 0.994i)11-s + (0.104 + 0.994i)12-s + (−0.438 − 0.898i)13-s + (0.997 − 0.0697i)14-s + (−0.719 − 0.694i)16-s + (−0.615 + 0.788i)17-s − 18-s + (−0.0348 + 0.999i)21-s + (0.882 − 0.469i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.856 - 0.516i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.856 - 0.516i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ -0.856 - 0.516i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3138742987 - 1.129015786i\)
\(L(\frac12)\) \(\approx\) \(0.3138742987 - 1.129015786i\)
\(L(1)\) \(\approx\) \(0.7978181984 - 0.4343382832i\)
\(L(1)\) \(\approx\) \(0.7978181984 - 0.4343382832i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.559 - 0.829i)T \)
3 \( 1 + (0.882 - 0.469i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (-0.438 - 0.898i)T \)
17 \( 1 + (-0.615 + 0.788i)T \)
23 \( 1 + (-0.241 - 0.970i)T \)
29 \( 1 + (0.615 + 0.788i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (0.719 + 0.694i)T \)
43 \( 1 + (0.766 - 0.642i)T \)
47 \( 1 + (-0.615 - 0.788i)T \)
53 \( 1 + (0.374 - 0.927i)T \)
59 \( 1 + (-0.961 + 0.275i)T \)
61 \( 1 + (-0.241 - 0.970i)T \)
67 \( 1 + (-0.0348 - 0.999i)T \)
71 \( 1 + (-0.848 - 0.529i)T \)
73 \( 1 + (0.438 - 0.898i)T \)
79 \( 1 + (0.882 - 0.469i)T \)
83 \( 1 + (0.669 + 0.743i)T \)
89 \( 1 + (0.719 - 0.694i)T \)
97 \( 1 + (-0.0348 + 0.999i)T \)
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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

−24.17483631522653413677468273454, −23.35645810798882642993832274724, −22.288555886474261104935572867888, −21.421871390681710277481917641715, −20.30607794904402903390302921245, −19.48419309281057292687840398377, −19.08088916942965353825578235160, −17.91332142382890770733505644459, −16.82929883007418968383250539556, −16.15810755637291326223098901536, −15.60212722703108415571296033365, −14.34872206777544870595906416595, −13.88525375570346553258639233965, −13.169626063206356162237066725974, −11.34635202103173375892981544827, −10.443961915364673878371364308740, −9.51165859286909346267879279314, −8.98933198633460755187418864811, −7.84431151600161226357247674305, −7.17398188531886593954944962498, −6.115937535595731164754118532881, −4.77998049814028113486603856719, −3.93893586800423820217992747796, −2.62728280670854928995427347943, −1.12105965920050280433568305130, 0.34955127390351064549701816820, 1.91811974830436373480006719782, 2.52122327794964450071986019772, 3.52488904992931693890617298830, 4.69530252133820078316343519888, 6.32429424386040827202334494220, 7.44706122995843915776987539236, 8.25864503922776693054279539494, 9.11760381845300643168944462090, 9.82482600063707173372116616411, 10.772162791637825161865375940362, 12.24797474941948064731796766738, 12.5838613852364964268152915828, 13.300974794342238661398527732326, 14.658559464226672555963288438678, 15.32485379735523868982622492436, 16.48113494035318108211509906572, 17.84444690767422184248676743862, 18.11958378690156967566996526167, 19.15823881296573114478255264921, 19.86707954816924032535305180887, 20.370504489205421416876204912827, 21.41442894052167797606791716947, 22.191004592229166941560833375816, 23.05503497689882827667853688866

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