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Properties

Label 1-111-111.2-r0-0-0
Degree $1$
Conductor $111$
Sign $0.934 + 0.355i$
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  + (−0.984 − 0.173i)2-s + (0.939 + 0.342i)4-s + (0.642 + 0.766i)5-s + (0.766 − 0.642i)7-s + (−0.866 − 0.5i)8-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.342 + 0.939i)13-s + (−0.866 + 0.5i)14-s + (0.766 + 0.642i)16-s + (−0.342 − 0.939i)17-s + (0.984 − 0.173i)19-s + (0.342 + 0.939i)20-s + (0.642 − 0.766i)22-s + (0.866 − 0.5i)23-s + ⋯
L(s)  = 1  + (−0.984 − 0.173i)2-s + (0.939 + 0.342i)4-s + (0.642 + 0.766i)5-s + (0.766 − 0.642i)7-s + (−0.866 − 0.5i)8-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (−0.342 + 0.939i)13-s + (−0.866 + 0.5i)14-s + (0.766 + 0.642i)16-s + (−0.342 − 0.939i)17-s + (0.984 − 0.173i)19-s + (0.342 + 0.939i)20-s + (0.642 − 0.766i)22-s + (0.866 − 0.5i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $0.934 + 0.355i$
Analytic conductor: \(0.515481\)
Root analytic conductor: \(0.515481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (2, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 111,\ (0:\ ),\ 0.934 + 0.355i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7798372014 + 0.1433007768i\)
\(L(\frac12)\) \(\approx\) \(0.7798372014 + 0.1433007768i\)
\(L(1)\) \(\approx\) \(0.8085682264 + 0.06292314655i\)
\(L(1)\) \(\approx\) \(0.8085682264 + 0.06292314655i\)

Euler product

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

−29.06631491683468333600946905946, −28.34090314479092905651255977224, −27.400569607693931125125482697954, −26.46292385515160346340758230135, −25.13947356024406709464067643357, −24.63891313540247987811649924634, −23.730310698051407918177393724406, −21.80877463129207535923830049216, −20.98633087947830010293584317362, −20.04513498462445303629344530877, −18.79377218753076268778750886014, −17.7879536694842373909049221279, −17.07610709642122220193464434191, −15.87047987808265170995197742336, −14.90556135858502907135755472407, −13.40951983719946088311248855404, −12.07329805247250960192036771645, −10.893977180664005191251102588479, −9.72363848637911072189805664913, −8.58389394533551996746499500222, −7.866438788124575028051487487999, −6.007652405766742567380460224915, −5.19282017914823457886283161235, −2.70880481319717501557452606604, −1.26028307963652550277133521912, 1.6464373788696772900984119986, 2.8944164257220388314972869801, 4.9126947965591876889641358361, 6.84134765537269193163907203242, 7.38626266110882323186309693794, 9.0038820458836038545992198250, 10.08170868597462998235344816377, 10.93656822997744357269613335810, 12.03822264480348451396514629388, 13.72160293189984965728297345219, 14.75848251387192823932257114471, 16.054821303754114952616352879157, 17.34037096346620856747819471796, 17.95500539479745191336192411815, 18.88135963141664319240356408235, 20.23934747794143055900203206122, 20.97557595982814824457854120326, 22.09348520909649813769464733661, 23.48764719904975708712890124656, 24.70857981520373786488905870838, 25.64352816676154203542722854785, 26.68390356807878780765536394311, 27.1375645491966111672194612028, 28.74212108496694315197923679087, 29.161457459560899106000631051975

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