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Properties

Label 1-475-475.146-r1-0-0
Degree $1$
Conductor $475$
Sign $-0.535 + 0.844i$
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  + (0.719 − 0.694i)2-s + (0.374 − 0.927i)3-s + (0.0348 − 0.999i)4-s + (−0.374 − 0.927i)6-s + (−0.5 − 0.866i)7-s + (−0.669 − 0.743i)8-s + (−0.719 − 0.694i)9-s + (0.913 − 0.406i)11-s + (−0.913 − 0.406i)12-s + (0.241 − 0.970i)13-s + (−0.961 − 0.275i)14-s + (−0.997 − 0.0697i)16-s + (−0.882 − 0.469i)17-s − 18-s + (−0.990 + 0.139i)21-s + (0.374 − 0.927i)22-s + ⋯
L(s)  = 1  + (0.719 − 0.694i)2-s + (0.374 − 0.927i)3-s + (0.0348 − 0.999i)4-s + (−0.374 − 0.927i)6-s + (−0.5 − 0.866i)7-s + (−0.669 − 0.743i)8-s + (−0.719 − 0.694i)9-s + (0.913 − 0.406i)11-s + (−0.913 − 0.406i)12-s + (0.241 − 0.970i)13-s + (−0.961 − 0.275i)14-s + (−0.997 − 0.0697i)16-s + (−0.882 − 0.469i)17-s − 18-s + (−0.990 + 0.139i)21-s + (0.374 − 0.927i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.535 + 0.844i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (146, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ -0.535 + 0.844i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-1.316495458 - 2.392846240i\)
\(L(\frac12)\) \(\approx\) \(-1.316495458 - 2.392846240i\)
\(L(1)\) \(\approx\) \(0.7471974837 - 1.417516243i\)
\(L(1)\) \(\approx\) \(0.7471974837 - 1.417516243i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (0.719 - 0.694i)T \)
3 \( 1 + (0.374 - 0.927i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.913 - 0.406i)T \)
13 \( 1 + (0.241 - 0.970i)T \)
17 \( 1 + (-0.882 - 0.469i)T \)
23 \( 1 + (0.559 + 0.829i)T \)
29 \( 1 + (0.882 - 0.469i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (0.809 - 0.587i)T \)
41 \( 1 + (0.997 + 0.0697i)T \)
43 \( 1 + (-0.939 + 0.342i)T \)
47 \( 1 + (-0.882 + 0.469i)T \)
53 \( 1 + (-0.0348 + 0.999i)T \)
59 \( 1 + (-0.438 - 0.898i)T \)
61 \( 1 + (0.559 + 0.829i)T \)
67 \( 1 + (-0.990 - 0.139i)T \)
71 \( 1 + (0.615 + 0.788i)T \)
73 \( 1 + (-0.241 - 0.970i)T \)
79 \( 1 + (0.374 - 0.927i)T \)
83 \( 1 + (-0.978 + 0.207i)T \)
89 \( 1 + (0.997 - 0.0697i)T \)
97 \( 1 + (-0.990 + 0.139i)T \)
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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

−24.33902735148228302180465542930, −23.08314645849990574562683049245, −22.42643354116670312626753302138, −21.70745777597821086704082552233, −21.17406261208710950379381296411, −20.07157084180686062893778028077, −19.246061010175956287954237536376, −17.96595481778829687360321497326, −16.89802661320047450854861865304, −16.27297947031209007440907410922, −15.426493841980440276005942415769, −14.800197035833899855421905859695, −14.03671196569386210868039283029, −13.034105473860462323337783035757, −12.00685620786903667341310355943, −11.229701419880403819854712478325, −9.81467352445784265389679353286, −8.87509865023669641210532324268, −8.41442896083713003932699691043, −6.77849692581236845514672703571, −6.20465387264779908845895027494, −4.88700626795363093028398386324, −4.24573045613270863752293759025, −3.193653173451713443938042552897, −2.18893212903873050826453160509, 0.53645004972819366721150625702, 1.27558737183579900922666174850, 2.7027891414435170922433104568, 3.44127506662503142745046389044, 4.52177382578359006497860097509, 5.99957309854030891720192093290, 6.6008240245174223212532050551, 7.66911883399822352994561078548, 8.94883857221476175372356301133, 9.84205056609149009187089815487, 11.01379883706757186108040618740, 11.721713382806857344545513938756, 12.79235073098122067827140558603, 13.4116587294079482083986700909, 13.99040088889745969856056108020, 14.948860639974700472226141209690, 15.95794354900435646297347862337, 17.3023612204820348999372691248, 18.0464961486357362798818384460, 19.22688233318895555960384534951, 19.68712669325670162492547288601, 20.28732071879385625652443416017, 21.25227364469005807074616071227, 22.443961677899068939605150616757, 23.001954753733306256912498216823

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