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Properties

Label 1-111-111.89-r0-0-0
Degree $1$
Conductor $111$
Sign $-0.464 - 0.885i$
Analytic cond. $0.515481$
Root an. cond. $0.515481$
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  + (0.642 − 0.766i)2-s + (−0.173 − 0.984i)4-s + (0.342 − 0.939i)5-s + (−0.939 − 0.342i)7-s + (−0.866 − 0.5i)8-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.984 − 0.173i)13-s + (−0.866 + 0.5i)14-s + (−0.939 + 0.342i)16-s + (0.984 + 0.173i)17-s + (−0.642 − 0.766i)19-s + (−0.984 − 0.173i)20-s + (0.342 + 0.939i)22-s + (0.866 − 0.5i)23-s + ⋯
L(s)  = 1  + (0.642 − 0.766i)2-s + (−0.173 − 0.984i)4-s + (0.342 − 0.939i)5-s + (−0.939 − 0.342i)7-s + (−0.866 − 0.5i)8-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.984 − 0.173i)13-s + (−0.866 + 0.5i)14-s + (−0.939 + 0.342i)16-s + (0.984 + 0.173i)17-s + (−0.642 − 0.766i)19-s + (−0.984 − 0.173i)20-s + (0.342 + 0.939i)22-s + (0.866 − 0.5i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $-0.464 - 0.885i$
Analytic conductor: \(0.515481\)
Root analytic conductor: \(0.515481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 111,\ (0:\ ),\ -0.464 - 0.885i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6758416868 - 1.117881813i\)
\(L(\frac12)\) \(\approx\) \(0.6758416868 - 1.117881813i\)
\(L(1)\) \(\approx\) \(1.019690603 - 0.8160879428i\)
\(L(1)\) \(\approx\) \(1.019690603 - 0.8160879428i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
37 \( 1 \)
good2 \( 1 + (0.642 - 0.766i)T \)
5 \( 1 + (0.342 - 0.939i)T \)
7 \( 1 + (-0.939 - 0.342i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (0.984 - 0.173i)T \)
17 \( 1 + (0.984 + 0.173i)T \)
19 \( 1 + (-0.642 - 0.766i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (0.866 + 0.5i)T \)
31 \( 1 + iT \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 - iT \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (0.939 - 0.342i)T \)
59 \( 1 + (-0.342 - 0.939i)T \)
61 \( 1 + (-0.984 + 0.173i)T \)
67 \( 1 + (0.939 + 0.342i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 - T \)
79 \( 1 + (-0.342 + 0.939i)T \)
83 \( 1 + (-0.173 + 0.984i)T \)
89 \( 1 + (0.342 + 0.939i)T \)
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

−29.82617570684338647833708311217, −29.14779340512032245041100666174, −27.4773569458315308147707437208, −26.28693844054171146455886839804, −25.70596694145591974416699330479, −24.84837534358111094761068622996, −23.30449207786046416897620664516, −22.89479821518036751579789273379, −21.64236396465830427188813040651, −21.03929270899617152787945708642, −19.034267498733526267059474190639, −18.3965369134507832831031311538, −16.94735632852782413267499263227, −15.98849099009566480936424421462, −15.040711617246579343399084442645, −13.86916328722721331400968401296, −13.122289438086927811958844732896, −11.75101685157455486233434518718, −10.392240155579355875046553775148, −8.93574568635934678186368050218, −7.607591558075890610450188653205, −6.28793213585456163900534447441, −5.720065349404640558043073827243, −3.7012332106416503053047628090, −2.79678588064394895149390756875, 1.171510158759399718803548661639, 2.86869010281102853206170058020, 4.28345031344728435540143927472, 5.42141263260524557403469017359, 6.69867796249142527564666990477, 8.69050066945790253419347318594, 9.81895863735654959659765434168, 10.73117706653836160688065845717, 12.40294371719106970852966125133, 12.90145285918337856123536836548, 13.87970103126866489643935955624, 15.34436545353739683660884928187, 16.3678651940966131107354830609, 17.73731665724978824004089666853, 19.02153827652765669468610672678, 20.05671415300269933410981578313, 20.81526232534080430167495549385, 21.72977993738743857960215153183, 23.18222470294665388526348509562, 23.47771288860960138597764303377, 25.01931427079757678550333277110, 25.85546660528239133774665157071, 27.51845560562923494335503640692, 28.45848936970113166157069802089, 28.98561392999534061936079882462

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