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Properties

Label 1-2169-2169.419-r0-0-0
Degree $1$
Conductor $2169$
Sign $0.963 - 0.267i$
Analytic cond. $10.0727$
Root an. cond. $10.0727$
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  + (0.258 − 0.965i)2-s + (−0.866 − 0.5i)4-s + (0.965 + 0.258i)5-s + (−0.608 + 0.793i)7-s + (−0.707 + 0.707i)8-s + (0.5 − 0.866i)10-s + (−0.608 + 0.793i)11-s + (−0.608 − 0.793i)13-s + (0.608 + 0.793i)14-s + (0.5 + 0.866i)16-s + (0.382 − 0.923i)17-s + (0.130 − 0.991i)19-s + (−0.707 − 0.707i)20-s + (0.608 + 0.793i)22-s + (−0.608 + 0.793i)23-s + ⋯
L(s)  = 1  + (0.258 − 0.965i)2-s + (−0.866 − 0.5i)4-s + (0.965 + 0.258i)5-s + (−0.608 + 0.793i)7-s + (−0.707 + 0.707i)8-s + (0.5 − 0.866i)10-s + (−0.608 + 0.793i)11-s + (−0.608 − 0.793i)13-s + (0.608 + 0.793i)14-s + (0.5 + 0.866i)16-s + (0.382 − 0.923i)17-s + (0.130 − 0.991i)19-s + (−0.707 − 0.707i)20-s + (0.608 + 0.793i)22-s + (−0.608 + 0.793i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2169 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(2169\)    =    \(3^{2} \cdot 241\)
Sign: $0.963 - 0.267i$
Analytic conductor: \(10.0727\)
Root analytic conductor: \(10.0727\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2169} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 2169,\ (0:\ ),\ 0.963 - 0.267i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.516910440 - 0.2064300810i\)
\(L(\frac12)\) \(\approx\) \(1.516910440 - 0.2064300810i\)
\(L(1)\) \(\approx\) \(1.062551378 - 0.3573803430i\)
\(L(1)\) \(\approx\) \(1.062551378 - 0.3573803430i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
241 \( 1 \)
good2 \( 1 + (0.258 - 0.965i)T \)
5 \( 1 + (0.965 + 0.258i)T \)
7 \( 1 + (-0.608 + 0.793i)T \)
11 \( 1 + (-0.608 + 0.793i)T \)
13 \( 1 + (-0.608 - 0.793i)T \)
17 \( 1 + (0.382 - 0.923i)T \)
19 \( 1 + (0.130 - 0.991i)T \)
23 \( 1 + (-0.608 + 0.793i)T \)
29 \( 1 + (0.965 + 0.258i)T \)
31 \( 1 + (0.382 - 0.923i)T \)
37 \( 1 + (-0.991 - 0.130i)T \)
41 \( 1 + (-0.258 + 0.965i)T \)
43 \( 1 + (0.793 - 0.608i)T \)
47 \( 1 + (0.258 + 0.965i)T \)
53 \( 1 + (-0.258 + 0.965i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (0.258 + 0.965i)T \)
67 \( 1 + (0.707 + 0.707i)T \)
71 \( 1 + (0.608 + 0.793i)T \)
73 \( 1 + (0.382 + 0.923i)T \)
79 \( 1 + (0.965 - 0.258i)T \)
83 \( 1 + (0.866 - 0.5i)T \)
89 \( 1 + (-0.608 - 0.793i)T \)
97 \( 1 + iT \)
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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

−19.62936252370194271561714493553, −18.96402207984844190910919834481, −18.18875394625158814332969972245, −17.386316013825395310936069671580, −16.74422793403279394409505746053, −16.39478066049269696778603327398, −15.6229533669461114993025673567, −14.49954987475245048449235494838, −13.90502138066846096747205128229, −13.65934645204891881449550797818, −12.4960206895762541039333825767, −12.30957237376567196328567920286, −10.614688231856239396521771357549, −10.14600661537607652617064938331, −9.397779198624488984061332660204, −8.47828463190666479989789255138, −7.91489904804152614199145674333, −6.77449662262624824845018226691, −6.3636951415471713158920658556, −5.56264328843907678924074686683, −4.78972429114104450735509979579, −3.8773172359781516609779526982, −3.08306853864181695650250224805, −1.86305791661544593233178460740, −0.57054261159584157324310529217, 0.906285911351640128669088751203, 2.20680848576782029072743765673, 2.61371966183203345265722917700, 3.27001222778039046469380584309, 4.659551831108803788290606632217, 5.283511425037983727079551930785, 5.874041688451432804360785747044, 6.88934930782073333881203143034, 7.897983051679214577460601048205, 9.03441551216686469651051559070, 9.64091290530839624054379638187, 10.03599758189055932681205364407, 10.83842687538198267319132209315, 11.84171926397063059874527272241, 12.43830192536495491309038352874, 13.10268610392137261056176655322, 13.72078625539687293104291150554, 14.48294733885807745286996243493, 15.348584984927775123991531663488, 15.86146382002822188302619439565, 17.35724079106948639849719010583, 17.64455404004860687837622913447, 18.42035426734770259384234548407, 18.96420328381795857235550665986, 19.90635254596695312328367142743

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