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Properties

Label 1-111-111.8-r0-0-0
Degree $1$
Conductor $111$
Sign $0.660 - 0.751i$
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  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.5 − 0.866i)7-s − i·8-s + 10-s + 11-s + (0.866 − 0.5i)13-s + i·14-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (0.866 − 0.5i)19-s + (−0.866 − 0.5i)20-s + (−0.866 − 0.5i)22-s − i·23-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)5-s + (−0.5 − 0.866i)7-s − i·8-s + 10-s + 11-s + (0.866 − 0.5i)13-s + i·14-s + (−0.5 + 0.866i)16-s + (0.866 + 0.5i)17-s + (0.866 − 0.5i)19-s + (−0.866 − 0.5i)20-s + (−0.866 − 0.5i)22-s − i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(111\)    =    \(3 \cdot 37\)
Sign: $0.660 - 0.751i$
Analytic conductor: \(0.515481\)
Root analytic conductor: \(0.515481\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{111} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 111,\ (0:\ ),\ 0.660 - 0.751i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5684304020 - 0.2572366731i\)
\(L(\frac12)\) \(\approx\) \(0.5684304020 - 0.2572366731i\)
\(L(1)\) \(\approx\) \(0.6500360196 - 0.1677609548i\)
\(L(1)\) \(\approx\) \(0.6500360196 - 0.1677609548i\)

Euler product

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

−29.20959983482436955171515615211, −28.25151077375228398012760778458, −27.61571331104620474941945861825, −26.66203493015929699299821969428, −25.35594440628681708252368319151, −24.806197213455072316650058302101, −23.59451760834280649247302558936, −22.72091248115746086000581473008, −21.108238622779739204990581400540, −19.911080735722701558235340462187, −19.10788390422501729321144596004, −18.26176831346771909827634631623, −16.79974872506722154764564604789, −16.05286125730881233581925146780, −15.22694329059216373843796223895, −13.91664364351943800799541240392, −12.08404583900962286551502108469, −11.45812116959389169236555464627, −9.6937674437313361711511555048, −8.89297404906821733405471079743, −7.82034153624179659212658148004, −6.52363081314068441063054671311, −5.30509178758140489510305563571, −3.47476126712780434319198192278, −1.35716483168152416834975152809, 1.00359692796775783765040698108, 3.17379044843813013497596431807, 3.96754664809303985024975337678, 6.50067199164577397302951766379, 7.46042360023421027058728697200, 8.58310784701891324158397621601, 9.96785692350301540673965467322, 10.90995869649934195453978091381, 11.87435968556038700862077434397, 13.07908202064269521305487679569, 14.59343631079928711887973787603, 15.991939234159099305498758386084, 16.73961760973232605913782277528, 18.0064043850287485311365406371, 19.043730326038378647932339439925, 19.86098704176846053045942538044, 20.62254920656115659786200530266, 22.17930914562166505346572794079, 22.950176355126939543135235647584, 24.28551591755904625453199095480, 25.694176698601989912741380539670, 26.360316425861754075343660116899, 27.38883636336814032516047561555, 28.04119228279920814747060319134, 29.37018699383587379479411871434

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