Properties

Label 1-475-475.147-r0-0-0
Degree 11
Conductor 475475
Sign 0.161+0.986i-0.161 + 0.986i
Analytic cond. 2.205892.20589
Root an. cond. 2.205892.20589
Motivic weight 00
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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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

Λ(s)=(475s/2ΓR(s)L(s)=((0.161+0.986i)Λ(1s)\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}
Λ(s)=(475s/2ΓR(s)L(s)=((0.161+0.986i)Λ(1s)\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: 11
Conductor: 475475    =    52195^{2} \cdot 19
Sign: 0.161+0.986i-0.161 + 0.986i
Analytic conductor: 2.205892.20589
Root analytic conductor: 2.205892.20589
Motivic weight: 00
Rational: no
Arithmetic: yes
Character: χ475(147,)\chi_{475} (147, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (1, 475, (0: ), 0.161+0.986i)(1,\ 475,\ (0:\ ),\ -0.161 + 0.986i)

Particular Values

L(12)L(\frac{1}{2}) \approx 1.096953537+1.291170169i1.096953537 + 1.291170169i
L(12)L(\frac12) \approx 1.096953537+1.291170169i1.096953537 + 1.291170169i
L(1)L(1) \approx 1.130317191+0.7192136359i1.130317191 + 0.7192136359i
L(1)L(1) \approx 1.130317191+0.7192136359i1.130317191 + 0.7192136359i

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1 1
19 1 1
good2 1+(0.0697+0.997i)T 1 + (0.0697 + 0.997i)T
3 1+(0.999+0.0348i)T 1 + (0.999 + 0.0348i)T
7 1+(0.8660.5i)T 1 + (-0.866 - 0.5i)T
11 1+(0.104+0.994i)T 1 + (-0.104 + 0.994i)T
13 1+(0.829+0.559i)T 1 + (0.829 + 0.559i)T
17 1+(0.9270.374i)T 1 + (0.927 - 0.374i)T
23 1+(0.694+0.719i)T 1 + (0.694 + 0.719i)T
29 1+(0.374+0.927i)T 1 + (-0.374 + 0.927i)T
31 1+(0.6690.743i)T 1 + (-0.669 - 0.743i)T
37 1+(0.587+0.809i)T 1 + (0.587 + 0.809i)T
41 1+(0.961+0.275i)T 1 + (-0.961 + 0.275i)T
43 1+(0.9840.173i)T 1 + (0.984 - 0.173i)T
47 1+(0.9270.374i)T 1 + (-0.927 - 0.374i)T
53 1+(0.139+0.990i)T 1 + (0.139 + 0.990i)T
59 1+(0.241+0.970i)T 1 + (-0.241 + 0.970i)T
61 1+(0.719+0.694i)T 1 + (-0.719 + 0.694i)T
67 1+(0.5290.848i)T 1 + (-0.529 - 0.848i)T
71 1+(0.8820.469i)T 1 + (0.882 - 0.469i)T
73 1+(0.8290.559i)T 1 + (0.829 - 0.559i)T
79 1+(0.03480.999i)T 1 + (0.0348 - 0.999i)T
83 1+(0.7430.669i)T 1 + (0.743 - 0.669i)T
89 1+(0.961+0.275i)T 1 + (0.961 + 0.275i)T
97 1+(0.5290.848i)T 1 + (0.529 - 0.848i)T
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   L(s)=p (1αpps)1L(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 ZZ-function along the critical line