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Properties

Label 1-140-140.27-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 $0$

Origins

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Normalization:  

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 + ⋯
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

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

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: $\chi_{140} (27, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 140,\ (1:\ ),\ -0.525 + 0.850i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.08216507969 + 0.1473716561i\)
\(L(\frac12)\) \(\approx\) \(0.08216507969 + 0.1473716561i\)
\(L(1)\) \(\approx\) \(0.7308946952 - 0.1725408324i\)
\(L(1)\) \(\approx\) \(0.7308946952 - 0.1725408324i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + T \)
11 \( 1 \)
13 \( 1 \)
17 \( 1 \)
19 \( 1 \)
23 \( 1 \)
29 \( 1 - T \)
31 \( 1 \)
37 \( 1 - T \)
41 \( 1 \)
43 \( 1 + iT \)
47 \( 1 \)
53 \( 1 \)
59 \( 1 \)
61 \( 1 - iT \)
67 \( 1 \)
71 \( 1 - T \)
73 \( 1 \)
79 \( 1 \)
83 \( 1 \)
89 \( 1 + iT \)
97 \( 1 \)
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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

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

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