# 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 $0$

# Related objects

## 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$$
show less
$$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