↑

Properties

Label 1-475-475.147-r0-0-0
Degree $1$
Conductor $475$
Sign $-0.161 + 0.986i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.0697 + 0.997i)2-s + (0.999 + 0.0348i)3-s + (−0.990 + 0.139i)4-s + (0.0348 + 0.999i)6-s + (−0.866 − 0.5i)7-s + (−0.207 − 0.978i)8-s + (0.997 + 0.0697i)9-s + (−0.104 + 0.994i)11-s + (−0.994 + 0.104i)12-s + (0.829 + 0.559i)13-s + (0.438 − 0.898i)14-s + (0.961 − 0.275i)16-s + (0.927 − 0.374i)17-s + i·18-s + (−0.848 − 0.529i)21-s + (−0.999 − 0.0348i)22-s + ⋯
L(s)  = 1  + (0.0697 + 0.997i)2-s + (0.999 + 0.0348i)3-s + (−0.990 + 0.139i)4-s + (0.0348 + 0.999i)6-s + (−0.866 − 0.5i)7-s + (−0.207 − 0.978i)8-s + (0.997 + 0.0697i)9-s + (−0.104 + 0.994i)11-s + (−0.994 + 0.104i)12-s + (0.829 + 0.559i)13-s + (0.438 − 0.898i)14-s + (0.961 − 0.275i)16-s + (0.927 − 0.374i)17-s + i·18-s + (−0.848 − 0.529i)21-s + (−0.999 − 0.0348i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.161 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.161 + 0.986i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (147, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ -0.161 + 0.986i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.096953537 + 1.291170169i\)
\(L(\frac12)\) \(\approx\) \(1.096953537 + 1.291170169i\)
\(L(1)\) \(\approx\) \(1.130317191 + 0.7192136359i\)
\(L(1)\) \(\approx\) \(1.130317191 + 0.7192136359i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.0697 + 0.997i)T \)
3 \( 1 + (0.999 + 0.0348i)T \)
7 \( 1 + (-0.866 - 0.5i)T \)
11 \( 1 + (-0.104 + 0.994i)T \)
13 \( 1 + (0.829 + 0.559i)T \)
17 \( 1 + (0.927 - 0.374i)T \)
23 \( 1 + (0.694 + 0.719i)T \)
29 \( 1 + (-0.374 + 0.927i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (0.587 + 0.809i)T \)
41 \( 1 + (-0.961 + 0.275i)T \)
43 \( 1 + (0.984 - 0.173i)T \)
47 \( 1 + (-0.927 - 0.374i)T \)
53 \( 1 + (0.139 + 0.990i)T \)
59 \( 1 + (-0.241 + 0.970i)T \)
61 \( 1 + (-0.719 + 0.694i)T \)
67 \( 1 + (-0.529 - 0.848i)T \)
71 \( 1 + (0.882 - 0.469i)T \)
73 \( 1 + (0.829 - 0.559i)T \)
79 \( 1 + (0.0348 - 0.999i)T \)
83 \( 1 + (0.743 - 0.669i)T \)
89 \( 1 + (0.961 + 0.275i)T \)
97 \( 1 + (0.529 - 0.848i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−23.40001463373897584597967949723, −22.574613899820771668450917099661, −21.57087208807053361484811171938, −21.05127084949523791794161363389, −20.18580196232181493326232159620, −19.256304485363139891073937464583, −18.85891652768281244257115281853, −18.100107022570621037372934146106, −16.65817776342510119965856803127, −15.70554646601963378740609899704, −14.69566270660942364608636914594, −13.865276286008835628825478202175, −12.976360893175867710332972347320, −12.54904305282141392604302983335, −11.20798752694527655672332586345, −10.31233425175295733005224765726, −9.42595499414099065933080295292, −8.65431693992369876096621867161, −7.93245480777772725001994268051, −6.326026657459784854476358553461, −5.29096920756857327482676839857, −3.73680758594325288929287733147, −3.27886765680325856833246338566, −2.32281981631292686683463607840, −0.95446375228598719252223085859, 1.38919290145519021995607035842, 3.130467921307834510357211504011, 3.88640322060920661628689968933, 4.913851995248517395825410515832, 6.26943778196027955790400142953, 7.214943915393773630002696293468, 7.741354501363980914538052830177, 9.05912491581363157442727011753, 9.50815616112200526313678932961, 10.46471872791159582392878620660, 12.22971643209601874165710036581, 13.19943931177412626907037106590, 13.65411197315391053622915050440, 14.69168983930552415982467740473, 15.33181693600992988643339358427, 16.26563741054596999794398254665, 16.87590226426984214543757991407, 18.21982706090848766180445048934, 18.7776418550411220437186184601, 19.75354889824837637728984255571, 20.63913382154884925252338655983, 21.54205758469719294503860058154, 22.608846497967784864451669999293, 23.357389689890666435548896284054, 24.04815831093765428997144377552

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