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Properties

Label 1-511-511.132-r0-0-0
Degree $1$
Conductor $511$
Sign $-0.809 - 0.587i$
Analytic cond. $2.37307$
Root an. cond. $2.37307$
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  + (−0.939 − 0.342i)2-s + (0.866 − 0.5i)3-s + (0.766 + 0.642i)4-s + (−0.906 − 0.422i)5-s + (−0.984 + 0.173i)6-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)9-s + (0.707 + 0.707i)10-s + (0.422 − 0.906i)11-s + (0.984 + 0.173i)12-s + (−0.819 − 0.573i)13-s + (−0.996 + 0.0871i)15-s + (0.173 + 0.984i)16-s + (0.965 − 0.258i)17-s + (−0.766 + 0.642i)18-s + (0.342 − 0.939i)19-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.866 − 0.5i)3-s + (0.766 + 0.642i)4-s + (−0.906 − 0.422i)5-s + (−0.984 + 0.173i)6-s + (−0.5 − 0.866i)8-s + (0.5 − 0.866i)9-s + (0.707 + 0.707i)10-s + (0.422 − 0.906i)11-s + (0.984 + 0.173i)12-s + (−0.819 − 0.573i)13-s + (−0.996 + 0.0871i)15-s + (0.173 + 0.984i)16-s + (0.965 − 0.258i)17-s + (−0.766 + 0.642i)18-s + (0.342 − 0.939i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 511 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 511 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.809 - 0.587i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(511\)    =    \(7 \cdot 73\)
Sign: $-0.809 - 0.587i$
Analytic conductor: \(2.37307\)
Root analytic conductor: \(2.37307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{511} (132, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 511,\ (0:\ ),\ -0.809 - 0.587i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2644467371 - 0.8146099898i\)
\(L(\frac12)\) \(\approx\) \(0.2644467371 - 0.8146099898i\)
\(L(1)\) \(\approx\) \(0.6592113338 - 0.4339331780i\)
\(L(1)\) \(\approx\) \(0.6592113338 - 0.4339331780i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
73 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-0.906 - 0.422i)T \)
11 \( 1 + (0.422 - 0.906i)T \)
13 \( 1 + (-0.819 - 0.573i)T \)
17 \( 1 + (0.965 - 0.258i)T \)
19 \( 1 + (0.342 - 0.939i)T \)
23 \( 1 + (0.342 + 0.939i)T \)
29 \( 1 + (-0.906 + 0.422i)T \)
31 \( 1 + (-0.0871 + 0.996i)T \)
37 \( 1 + (-0.939 + 0.342i)T \)
41 \( 1 + (0.173 - 0.984i)T \)
43 \( 1 + (-0.965 - 0.258i)T \)
47 \( 1 + (-0.573 - 0.819i)T \)
53 \( 1 + (-0.422 - 0.906i)T \)
59 \( 1 + (0.573 - 0.819i)T \)
61 \( 1 + (-0.984 - 0.173i)T \)
67 \( 1 + (-0.984 + 0.173i)T \)
71 \( 1 + (0.939 + 0.342i)T \)
79 \( 1 + (0.984 - 0.173i)T \)
83 \( 1 + (0.707 + 0.707i)T \)
89 \( 1 + (-0.173 - 0.984i)T \)
97 \( 1 + (-0.866 - 0.5i)T \)
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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

−24.27453836270922523779995798281, −23.11927241840612521894537269861, −22.344804476164326454997469390153, −21.00850279523457638393216996269, −20.384326459750933463585343840536, −19.53132595254468960616920027213, −18.99969696004993761927785337298, −18.24861087997193745688925519646, −16.80413691458267473045946974749, −16.439555748886541281290899019792, −15.200639689025573160799177082733, −14.83425432764670562799236352142, −14.178169179368381802582115516762, −12.47609554854533247413361477810, −11.64198811997708850448205593009, −10.520995011915675107257234276226, −9.80554785910649149356350441341, −9.057694883998552274174160991655, −7.83049844696267975082545607171, −7.5768477630575312307254387286, −6.451806385476630140136684418583, −4.92044519240891260834383816136, −3.87115711848352919729912545628, −2.727800439823976023672661972323, −1.6369232820422940449947047375, 0.57420012354848732805848155851, 1.66727152788562004099676731032, 3.19265080228431652878365881905, 3.482282326673978645850704199632, 5.22813931394462313344486616028, 6.90182402863937053208450138835, 7.48199615893174141497975094067, 8.34359962931010499593711408056, 9.00483139176315329943096115062, 9.87312338544799657937971981722, 11.14289515887366139198705015276, 11.971403708375380864301973575386, 12.6438586784263770309288699073, 13.6743136086671373787799021101, 14.88128180306085038913674518853, 15.62405826138448792500760112600, 16.5280145993997346183311328330, 17.42597447781350916487623989861, 18.42416760148929044823295042137, 19.308300528631184454198940114006, 19.60120691087918718613395757771, 20.4127437282366431182853241047, 21.21443690245250931792850208309, 22.21682269639070633950804188949, 23.61430597537381313834748605190

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