L-function 1-140-140.83-r1-0-0

• Label: 1-140-140.83-r1-0-0
• Degree: $1$
• Conductor: $140$
• Sign: $-0.525 - 0.850i$
• Analytic cond.: $15.0450$
• Root an. cond.: $15.0450$
• Motivic weight: $0$
• Arithmetic: yes
• Rational: no
• Primitive: yes
• Self-dual: no
• Analytic rank: $0$

Dirichlet series

L(s)  = 1

+ i·3^-s − 9^-s − 11^-s − i·13^-s + i·17^-s − 19^-s − i·23^-s − i·27^-s − 29^-s + 31^-s − i·33^-s − i·37^-s + 39^-s − 41^-s − i·43^-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree:	\(1\)
Conductor:	\(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign:	$-0.525 - 0.850i$
Analytic conductor:	\(15.0450\)
Root analytic conductor:	\(15.0450\)
Motivic weight:	\(0\)
Rational:	no
Arithmetic:	yes
Character:	$\chi_{140} (83, \cdot )$
Primitive:	yes
Self-dual:	no
Analytic rank:	\(0\)
Selberg data:	\((1,\ 140,\ (1:\ ),\ -0.525 - 0.850i)\)

Particular Values

\(L(\frac{1}{2})\)	\(\approx\)	\(0.08216507969 - 0.1473716561i\)
\(L(1)\)	\(\approx\)	\(0.7308946952 + 0.1725408324i\)

Euler product

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

	$p$	$F_p(T)$
bad	2	\( 1 \)
	5	\( 1 \)
	7	\( 1 \)
good	3	\( 1 + T \)
	11	\( 1 \)
	13	\( 1 \)
	17	\( 1 \)
	19	\( 1 \)
	23	\( 1 \)
	29	\( 1 - T \)
	31	\( 1 \)
	37	\( 1 - T \)

Imaginary part of the first few zeros on the critical line

−28.76738169154380200076842074293, −27.60007362489503159840821926516, −26.24066042064482119688548467403, −25.5504188781249847889456145745, −24.388591952752209945783586035100, −23.63794279840082816392030849251, −22.80888008568049079738602624620, −21.42685224225089977238904525265, −20.42873927102153000711367381433, −19.19991809570454315614611201224, −18.55459108576250076056868534428, −17.51092979691676403023657779386, −16.454926290257718422393752321, −15.155394046424681172907188940682, −13.87184128600035893493995925144, −13.16254613335902274987357913570, −11.97855823764077818994494660054, −11.06227973309800953059738805119, −9.52282482429614426950055490158, −8.25546793246418115848227508292, −7.253062983615418294430564189041, −6.16838580120941914467066301067, −4.81906831047126270424878719672, −2.962013405674335888482472672290, −1.7003000440856520411801028682, 0.06148958587469873920054335907, 2.48225265517008762185380177372, 3.803159652038460673548006742095, 5.04337221010898316341564980984, 6.10901154784052707221180540269, 7.91654211328002122350467123513, 8.84387859434234904056202302715, 10.3629492920566380153931357616, 10.68520404598161463923603323644, 12.30735985633199354506880650636, 13.42300090109785530163628396405, 14.882994618696397530911061803679, 15.42115142164895548648950450918, 16.61510493718233581284164489711, 17.50595322560601303268532008324, 18.78560263346815692521214571614, 20.0459864270281619808906665715, 20.88142838653315308451977286445, 21.74881874210423433546664241039, 22.77412403462972807971077821973, 23.65903971624991941063525760431, 25.03710045163192403107719164916, 26.06834246947710459012358753750, 26.733136591344421097384292886460, 27.93400859184728525997654700311

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