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Properties

Label 1-475-475.14-r1-0-0
Degree $1$
Conductor $475$
Sign $-0.343 + 0.939i$
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  + (−0.374 − 0.927i)2-s + (0.559 + 0.829i)3-s + (−0.719 + 0.694i)4-s + (0.559 − 0.829i)6-s + (0.5 − 0.866i)7-s + (0.913 + 0.406i)8-s + (−0.374 + 0.927i)9-s + (−0.978 + 0.207i)11-s + (−0.978 − 0.207i)12-s + (−0.615 − 0.788i)13-s + (−0.990 − 0.139i)14-s + (0.0348 − 0.999i)16-s + (0.241 − 0.970i)17-s + 18-s + (0.997 − 0.0697i)21-s + (0.559 + 0.829i)22-s + ⋯
L(s)  = 1  + (−0.374 − 0.927i)2-s + (0.559 + 0.829i)3-s + (−0.719 + 0.694i)4-s + (0.559 − 0.829i)6-s + (0.5 − 0.866i)7-s + (0.913 + 0.406i)8-s + (−0.374 + 0.927i)9-s + (−0.978 + 0.207i)11-s + (−0.978 − 0.207i)12-s + (−0.615 − 0.788i)13-s + (−0.990 − 0.139i)14-s + (0.0348 − 0.999i)16-s + (0.241 − 0.970i)17-s + 18-s + (0.997 − 0.0697i)21-s + (0.559 + 0.829i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.343 + 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $-0.343 + 0.939i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (14, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ -0.343 + 0.939i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3727831226 + 0.5333677658i\)
\(L(\frac12)\) \(\approx\) \(0.3727831226 + 0.5333677658i\)
\(L(1)\) \(\approx\) \(0.8347615750 - 0.08511162977i\)
\(L(1)\) \(\approx\) \(0.8347615750 - 0.08511162977i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.374 - 0.927i)T \)
3 \( 1 + (0.559 + 0.829i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.978 + 0.207i)T \)
13 \( 1 + (-0.615 - 0.788i)T \)
17 \( 1 + (0.241 - 0.970i)T \)
23 \( 1 + (0.882 + 0.469i)T \)
29 \( 1 + (0.241 + 0.970i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
41 \( 1 + (-0.0348 + 0.999i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (0.241 + 0.970i)T \)
53 \( 1 + (-0.719 + 0.694i)T \)
59 \( 1 + (-0.848 - 0.529i)T \)
61 \( 1 + (-0.882 - 0.469i)T \)
67 \( 1 + (-0.997 - 0.0697i)T \)
71 \( 1 + (-0.438 + 0.898i)T \)
73 \( 1 + (0.615 - 0.788i)T \)
79 \( 1 + (-0.559 - 0.829i)T \)
83 \( 1 + (0.104 + 0.994i)T \)
89 \( 1 + (-0.0348 - 0.999i)T \)
97 \( 1 + (-0.997 + 0.0697i)T \)
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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

−23.71466287528256483153873972518, −22.869139279148931287578562240242, −21.609651738768223478055434414499, −20.83235050607196732915858293135, −19.44435465768118405925613081244, −18.95889183315149720160453166753, −18.25853029247078339083987359137, −17.458358879029304899603672363, −16.56580534627789204848403253090, −15.2930483650965251906985861614, −14.88672369643123486893850267286, −13.960556419654381261847942820270, −13.07543416934458086679530731341, −12.21181235054152116001433658929, −10.98467782784841596423408549443, −9.69408606750748198781031523598, −8.8180092661197340926048579429, −8.0965327317870373723053567906, −7.38617536040975337988008201128, −6.283956532173624986503009258057, −5.500110149061690043324872735438, −4.3008130062647142832228657039, −2.63704485496949478708120750234, −1.658580706339879118548313549924, −0.18093549363904279009746790618, 1.28180703815642606217041516971, 2.74582579807025822841917216467, 3.30239437967594671224181709032, 4.750512397216102441553597497104, 5.028198512305032775511202569317, 7.387474012141561750112072479592, 7.927631335834720754368875612454, 8.99496515334679857405136603710, 9.94898433146783628988162615160, 10.54140850021035938080327988544, 11.24718515335488568714341861536, 12.50514150178540376653074732724, 13.47545156712969466409364247496, 14.15913569119874598735452374765, 15.200399033920929419058062959535, 16.250388072008947384894129221235, 17.12518907352116942422003379914, 17.93526237708684098124041241872, 18.91848281119624205469614325139, 20.0142646101578822405930492260, 20.35724982568369897593877652946, 21.11395110270383343345783600694, 21.87316846467471990738907313546, 22.83403873019976352287406656510, 23.56903123613072162012892405616

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