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Properties

Label 1-1017-1017.677-r1-0-0
Degree $1$
Conductor $1017$
Sign $0.984 - 0.173i$
Analytic cond. $109.291$
Root an. cond. $109.291$
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  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s − 8-s − 10-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + 17-s − 19-s + (−0.5 − 0.866i)20-s + (−0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)4-s + (−0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s − 8-s − 10-s + (0.5 + 0.866i)11-s + (−0.5 + 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.5 − 0.866i)16-s + 17-s − 19-s + (−0.5 − 0.866i)20-s + (−0.5 + 0.866i)22-s + (−0.5 + 0.866i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1017 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1017 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1017\)    =    \(3^{2} \cdot 113\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(109.291\)
Root analytic conductor: \(109.291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1017} (677, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1017,\ (1:\ ),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4329925635 - 0.03788194070i\)
\(L(\frac12)\) \(\approx\) \(0.4329925635 - 0.03788194070i\)
\(L(1)\) \(\approx\) \(0.7006095261 + 0.5695040438i\)
\(L(1)\) \(\approx\) \(0.7006095261 + 0.5695040438i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
113 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + T \)
19 \( 1 - T \)
23 \( 1 + (-0.5 + 0.866i)T \)
29 \( 1 + (-0.5 - 0.866i)T \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 - T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.5 + 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (0.5 - 0.866i)T \)
71 \( 1 + T \)
73 \( 1 - T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + T \)
97 \( 1 + (-0.5 - 0.866i)T \)
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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

−21.45941554592748720210746593574, −20.616217607940453776964911194643, −19.99357592334872086625202380382, −19.01764607419045264174585734890, −18.90025825156750253240276265287, −17.61494040466221308223127026232, −16.63153339830833982350754333515, −15.893096760029343743869916160187, −14.952891213582027441819536728991, −14.35704547571606744110808804080, −13.140976895606009989293840167946, −12.5772476135044312731214382036, −12.10559052315898377997064851323, −11.20821148327421744738459315457, −10.30022597990694722102588841646, −9.32606391532441709082776171155, −8.73012814735868834468119814798, −7.8602396699710150755677928031, −6.2825518219958222845998791502, −5.620424028203932333008770221827, −4.817475056398958051991039095341, −3.75143009899910990462029643209, −3.059792180591074444847942284659, −1.93976075551975009955911566825, −0.75611137771870506047126365822, 0.10122534111790533029252172863, 1.96220665680658062485940496168, 3.31860349459374168180613836895, 3.92597515240155768366592391301, 4.680450372964622107742912989420, 5.98690554618352985508632842619, 6.81500413280598409678334106209, 7.28843233037803265738930974092, 8.01431397631438790030798179374, 9.32528851813561618093731353249, 9.99468947769544890475289540914, 11.084340516282661377679031973137, 12.08805797740856071373564522794, 12.6335702830231810331991909223, 13.87255252992639554472838647336, 14.26391287803051620307664009366, 15.05555822888903978842868740546, 15.78114560830521197775912809235, 16.71121388562025107411367518794, 17.230385585993808801362946653, 18.09888036695631511057954311041, 19.15990455463133111332614038901, 19.621941037075235671833987170569, 20.81478578355188341241058208124, 21.62549118770797480279723471762

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