L-function 1-111-111.14-r0-0-0

• Label: 1-111-111.14-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: $0$

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 + ⋯

Functional equation

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

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:	$\chi_{111} (14, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 111,\ (0:\ ),\ 0.660 + 0.751i)\)

Particular Values

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

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)

	$p$	$F_p(T)$
bad	3	\( 1 \)
	37	\( 1 \)
good	2	\( 1 + (-0.866 + 0.5i)T \)
	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	\( 1 - iT \)

Imaginary part of the first few zeros on the critical line

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

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