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Properties

Label 1-475-475.109-r1-0-0
Degree $1$
Conductor $475$
Sign $0.915 + 0.401i$
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  + (0.848 + 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

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.915 + 0.401i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.915 + 0.401i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ 0.915 + 0.401i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.728676606 + 1.201939272i\)
\(L(\frac12)\) \(\approx\) \(5.728676606 + 1.201939272i\)
\(L(1)\) \(\approx\) \(2.672416444 + 0.4938280463i\)
\(L(1)\) \(\approx\) \(2.672416444 + 0.4938280463i\)

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

−23.50626390273736764671736745411, −22.41322936517352823838552536673, −21.82289263438578928615332901958, −20.97243034853066277515416328319, −20.2872326056008885660051059647, −19.62008431934650183474650840440, −18.68193557291014314079006441611, −17.85061650280421931517869755710, −16.2793484074532542856577251131, −15.38647649337820918378168290285, −14.71016889705990805709880031335, −14.238651613430989951808995145984, −13.036117254856870044670109044049, −12.348894476722078452697736661430, −11.46281991028633518720613354029, −10.18852349792285508267407906844, −9.64983610951562908603498148163, −8.45423883370652898192587310469, −7.51638146442187982626474841577, −6.157497365359405006512938066, −5.12791129116382886585523131387, −4.21239677664906551308613969071, −3.180440170922173927578857824076, −2.29273492029241911888929891774, −1.29850076489228791728741404169, 1.20613814278812533846606754714, 2.41203924731110323684991897633, 3.78179003564077954217824040457, 4.076937819584150811385219647851, 5.552499580982345722087118131802, 6.73689791516283641302496610051, 7.42370604331328289168320008441, 8.3057827229113109603377222219, 9.176664532633683575234012136053, 10.54253213878925957677175556016, 11.71195247243295751222445474064, 12.47546788757046143121814640441, 13.711846919783005241014211547055, 14.10485394018091289308629622445, 14.583014015974224294251840765102, 15.86884688643563276784373754485, 16.56331682743259614954943350903, 17.49224004215573589341960549212, 18.61543881966591630164653171508, 19.66167301385011679430706289489, 20.34005285715286807661856315575, 21.30816464808689600771946121082, 21.730703538921564223639425026892, 23.09591723001991167433159800384, 23.86813935485670239387200105692

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