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Properties

Label 1-140-140.103-r1-0-0
Degree $1$
Conductor $140$
Sign $-0.987 + 0.156i$
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  + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s − i·13-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s + (−0.866 + 0.5i)23-s − i·27-s − 29-s + (−0.5 + 0.866i)31-s + (−0.866 + 0.5i)33-s + (−0.866 + 0.5i)37-s + (−0.5 + 0.866i)39-s − 41-s − i·43-s + ⋯
L(s)  = 1  + (−0.866 − 0.5i)3-s + (0.5 + 0.866i)9-s + (0.5 − 0.866i)11-s − i·13-s + (−0.866 − 0.5i)17-s + (0.5 + 0.866i)19-s + (−0.866 + 0.5i)23-s − i·27-s − 29-s + (−0.5 + 0.866i)31-s + (−0.866 + 0.5i)33-s + (−0.866 + 0.5i)37-s + (−0.5 + 0.866i)39-s − 41-s − i·43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.987 + 0.156i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(140\)    =    \(2^{2} \cdot 5 \cdot 7\)
Sign: $-0.987 + 0.156i$
Analytic conductor: \(15.0450\)
Root analytic conductor: \(15.0450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{140} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 140,\ (1:\ ),\ -0.987 + 0.156i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02146861565 - 0.2734026913i\)
\(L(\frac12)\) \(\approx\) \(0.02146861565 - 0.2734026913i\)
\(L(1)\) \(\approx\) \(0.6351278243 - 0.1723909258i\)
\(L(1)\) \(\approx\) \(0.6351278243 - 0.1723909258i\)

Euler product

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

−28.353277411680243033124414060861, −28.02976858885865350858901392143, −26.646044544020452523588861761121, −26.01439579173487653221005803110, −24.4353913592242179040271937483, −23.72671656649746991161885864474, −22.48711300384586572560418353248, −21.9681859178448249058032693328, −20.78152347200445865247047203114, −19.76210528526917504159181004028, −18.36916658190101048097053659957, −17.46824513874773213532103058453, −16.58042509944460911294925399211, −15.54220011270948784904207335130, −14.56501064450467312328945995028, −13.09622009675722867149238252117, −11.926013494191647693042699931867, −11.13535359906085178432880405804, −9.879621086554415694393142102761, −8.990118255384604850100546795178, −7.161575702673779397040990742836, −6.2312823909370434527559760744, −4.81518940339006637444007588828, −3.91405632415784809844197690894, −1.83007186415757582590008896081, 0.11810271656582591673481217361, 1.62889464963376250305942040989, 3.4854977520942621465864353979, 5.15907824470954412325130075153, 6.08578934895098394890534968330, 7.27979933451325297016388623254, 8.463864227037763815793614535386, 10.022090559903222863578992758274, 11.12522423588817350127126676342, 12.001817758331519804289034028487, 13.124185507190038672450695279903, 14.087182760337545418381457230, 15.638188495854367613946857845619, 16.537070649642538382422661158982, 17.59133689151000041973161324693, 18.40358861857265107296378688679, 19.45986372868570561977808975517, 20.59055741020710910091723147127, 22.07959373836553349816810993882, 22.49098276427268118078074833044, 23.75459323098238418684666030273, 24.53215195575055796258460953096, 25.40813089170418552311995684340, 26.963294226461554834658957461933, 27.59869401075784240714729975929

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