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Properties

Label 1-280-280.269-r1-0-0
Degree $1$
Conductor $280$
Sign $-0.991 - 0.126i$
Analytic cond. $30.0901$
Root an. cond. $30.0901$
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  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s − 13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s − 27-s − 29-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)33-s + (−0.5 + 0.866i)37-s + (−0.5 − 0.866i)39-s − 41-s + 43-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)9-s + (0.5 + 0.866i)11-s − 13-s + (−0.5 − 0.866i)17-s + (−0.5 + 0.866i)19-s + (0.5 − 0.866i)23-s − 27-s − 29-s + (0.5 + 0.866i)31-s + (−0.5 + 0.866i)33-s + (−0.5 + 0.866i)37-s + (−0.5 − 0.866i)39-s − 41-s + 43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.991 - 0.126i$
Analytic conductor: \(30.0901\)
Root analytic conductor: \(30.0901\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (269, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 280,\ (1:\ ),\ -0.991 - 0.126i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.05809133048 + 0.9153980680i\)
\(L(\frac12)\) \(\approx\) \(-0.05809133048 + 0.9153980680i\)
\(L(1)\) \(\approx\) \(0.8900926342 + 0.4412431784i\)
\(L(1)\) \(\approx\) \(0.8900926342 + 0.4412431784i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 - T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (0.5 - 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + (-0.5 + 0.866i)T \)
41 \( 1 - T \)
43 \( 1 + T \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (-0.5 - 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (-0.5 + 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 - T \)
89 \( 1 + (0.5 - 0.866i)T \)
97 \( 1 + T \)
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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

−24.649899357276753228203404895896, −24.28787826705449313880734956206, −23.33771458380926883011573400004, −22.144817260763371103824255934535, −21.34374863720757308217484122130, −20.08374562484465478062522028501, −19.39957244573529177168554625704, −18.77964538758074522506340501682, −17.46669386101062912525272382708, −16.99529416361081118213507952258, −15.41731934247481556135886783129, −14.66054346398724133025998618980, −13.629358014346431835881629479208, −12.9102926552583969019974699333, −11.85887746257696272040672494006, −10.93451379125583950601786992542, −9.42868337257793086965315656607, −8.657697341386766024191850882342, −7.57606126520442148235359574283, −6.66482860569572772268266202, −5.61319982043608483209465767685, −4.033135561035002640097842999158, −2.8315516611928324403458473031, −1.689040306991070657461028764276, −0.24221412567922964586417384031, 1.93825044336528651684585608417, 3.081831659687347577762238109111, 4.3773023622470858244728447537, 5.066572148479931390939451589127, 6.64796639746106895673319619509, 7.75992022100376007413767636802, 8.92323197473606560065359803655, 9.72832669516286784371302185415, 10.56417662461236242238967073936, 11.76574184193099904597225922606, 12.78837169001649623377675017193, 14.10524014397834179450268494326, 14.75256375860875241466477594592, 15.58850765296363828919417853650, 16.67086506901344578322823772547, 17.3780782308637807030362613081, 18.715048646987580511028645173140, 19.695896687669003239006341240030, 20.46399277907121886182965781802, 21.19475209539868921238042000590, 22.446862680990665787641990247402, 22.68045598691516491723981732220, 24.26746314231267401290168520354, 25.11295118126900002540350765627, 25.82926342582816807885901503238

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