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Properties

Label 1-475-475.156-r0-0-0
Degree $1$
Conductor $475$
Sign $0.578 - 0.815i$
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

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Normalization:  

Dirichlet series

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

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.578 - 0.815i$
Analytic conductor: \(2.20589\)
Root analytic conductor: \(2.20589\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (156, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (0:\ ),\ 0.578 - 0.815i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4917902264 - 0.2540111847i\)
\(L(\frac12)\) \(\approx\) \(0.4917902264 - 0.2540111847i\)
\(L(1)\) \(\approx\) \(0.5996279384 + 0.02315869341i\)
\(L(1)\) \(\approx\) \(0.5996279384 + 0.02315869341i\)

Euler product

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

−24.334746302280881747634469263111, −23.15848342569931832240581842366, −22.49002494965091929869319728625, −21.11369668662218232388222721967, −20.15310677188422019825302196369, −19.614162096684094707703414070548, −18.6483744565686651218001875890, −17.95825230258796751638538476385, −17.59053174917706723381701319163, −16.219811277232579548960730805967, −15.54577144150055642549697575861, −14.60543009780335273829991316417, −13.3317040508418372490632941576, −12.38746892098013465851403769370, −11.81692193736259817063933275654, −10.715814763187573321911402289998, −9.591794907218694011323545608280, −8.81653852430512505070994884516, −7.88699037900819866545267396854, −7.06675746356265472526788279658, −6.19974835677442949635792891979, −5.29443226574013133784765000852, −3.1491555750456847459421128319, −2.34286989996757655300957301854, −1.237129869244644530995713968529, 0.44422001998303794986608191111, 2.15827398599618287795118012931, 3.48928590597269702743775752726, 4.14931657222776190563966235083, 5.877119782518320925647385179718, 6.54755758859782908539688799986, 7.936980140153856315443585577550, 8.71295693678647127275410623645, 9.56495458057744588491777731467, 10.47564380158001124397471025032, 10.97953381674603795919076473221, 11.9276357949190789720598401490, 13.37255754996006265094742466993, 14.29163770595289047737614564063, 15.4419026896992538963091313412, 16.16755987507131994890726855382, 16.7704707158814611586239230190, 17.43503696223830496657557319595, 18.74327881879076268791789183073, 19.431482101815253381707015318860, 20.243312304900594937945150621553, 21.04808079455502082241460936061, 21.69248404424336773926149573974, 22.76303924341615811809013889371, 23.81273340367661835526890688517

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