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Properties

Label 1-475-475.341-r1-0-0
Degree $1$
Conductor $475$
Sign $0.728 - 0.684i$
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  + (−0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)6-s + 7-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (−0.309 + 0.951i)12-s + (−0.309 + 0.951i)13-s + (−0.309 − 0.951i)14-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)17-s − 18-s + (0.809 − 0.587i)21-s + (0.809 − 0.587i)22-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)6-s + 7-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (0.309 + 0.951i)11-s + (−0.309 + 0.951i)12-s + (−0.309 + 0.951i)13-s + (−0.309 − 0.951i)14-s + (0.309 − 0.951i)16-s + (−0.809 − 0.587i)17-s − 18-s + (0.809 − 0.587i)21-s + (0.809 − 0.587i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.728 - 0.684i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(475\)    =    \(5^{2} \cdot 19\)
Sign: $0.728 - 0.684i$
Analytic conductor: \(51.0458\)
Root analytic conductor: \(51.0458\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{475} (341, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 475,\ (1:\ ),\ 0.728 - 0.684i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.255946592 - 0.8931924422i\)
\(L(\frac12)\) \(\approx\) \(2.255946592 - 0.8931924422i\)
\(L(1)\) \(\approx\) \(1.181872211 - 0.5561468477i\)
\(L(1)\) \(\approx\) \(1.181872211 - 0.5561468477i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 \)
19 \( 1 \)
good2 \( 1 + (-0.309 - 0.951i)T \)
3 \( 1 + (0.809 - 0.587i)T \)
7 \( 1 + T \)
11 \( 1 + (0.309 + 0.951i)T \)
13 \( 1 + (-0.309 + 0.951i)T \)
17 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.309 + 0.951i)T \)
41 \( 1 + (-0.309 + 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.809 + 0.587i)T \)
53 \( 1 + (0.809 - 0.587i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (0.309 + 0.951i)T \)
67 \( 1 + (0.809 + 0.587i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 + (0.309 + 0.951i)T \)
79 \( 1 + (0.809 - 0.587i)T \)
83 \( 1 + (-0.809 - 0.587i)T \)
89 \( 1 + (-0.309 - 0.951i)T \)
97 \( 1 + (0.809 - 0.587i)T \)
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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

−24.221287338044133077798448481419, −22.87779441234076145402974960821, −22.01867615342799171747685631536, −21.25950109650974484738826922869, −20.18617857972786530258946696897, −19.42651551613466713257435321918, −18.52659547518765066587814491729, −17.54305226562171837188646276339, −16.81574148689091066378694464078, −15.76697181023417185346222014646, −15.15679698479836152315420096401, −14.32538275502634987943023989300, −13.774041993822392659273902689268, −12.664180805507212641597804528851, −10.95982336922846269108071639021, −10.43037478177512736601300897020, −9.20040982389220244734776413649, −8.409799989186739761093083810347, −7.97994243670475504541100627546, −6.74619063509837604280869257116, −5.500018960898814269448369293841, −4.674926882549456824462475663160, −3.69572006379579188004675137762, −2.2380073638056415134616706483, −0.73338035817111409907603985933, 1.12106712886184222861401400652, 1.93342810841458579763198261238, 2.7836126015990060979147974271, 4.16460691503901832932441271768, 4.82961249938298601436817680101, 6.7658465134334638985597385936, 7.60466848020353811269323936481, 8.52962250123063921669261812930, 9.29609691606150841840458411375, 10.12567510341668104993170738696, 11.58279327523873590925686036061, 11.8698057914314517525604757843, 13.04013976865717165506839286382, 13.86598224354901349619052105114, 14.512279703216446491927127404832, 15.563267267218874039512539534524, 17.15653553869320127511473443971, 17.75527108115511894747482820140, 18.4515858078812717067635982108, 19.48225515966410232448120236052, 19.936520605276152073623720052643, 20.96512672120621361876529431489, 21.32972600552933002578299309070, 22.57908393376636400844256059180, 23.507427275370371666863597688176

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