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Properties

Label 1-987-987.755-r0-0-0
Degree $1$
Conductor $987$
Sign $0.0220 + 0.999i$
Analytic cond. $4.58360$
Root an. cond. $4.58360$
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  + (−0.460 + 0.887i)2-s + (−0.576 − 0.816i)4-s + (−0.917 + 0.398i)5-s + (0.990 − 0.136i)8-s + (0.0682 − 0.997i)10-s + (−0.962 − 0.269i)11-s + (−0.203 + 0.979i)13-s + (−0.334 + 0.942i)16-s + (0.962 − 0.269i)17-s + (0.917 + 0.398i)19-s + (0.854 + 0.519i)20-s + (0.682 − 0.730i)22-s + (−0.460 − 0.887i)23-s + (0.682 − 0.730i)25-s + (−0.775 − 0.631i)26-s + ⋯
L(s)  = 1  + (−0.460 + 0.887i)2-s + (−0.576 − 0.816i)4-s + (−0.917 + 0.398i)5-s + (0.990 − 0.136i)8-s + (0.0682 − 0.997i)10-s + (−0.962 − 0.269i)11-s + (−0.203 + 0.979i)13-s + (−0.334 + 0.942i)16-s + (0.962 − 0.269i)17-s + (0.917 + 0.398i)19-s + (0.854 + 0.519i)20-s + (0.682 − 0.730i)22-s + (−0.460 − 0.887i)23-s + (0.682 − 0.730i)25-s + (−0.775 − 0.631i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 987 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0220 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 987 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0220 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(987\)    =    \(3 \cdot 7 \cdot 47\)
Sign: $0.0220 + 0.999i$
Analytic conductor: \(4.58360\)
Root analytic conductor: \(4.58360\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{987} (755, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 987,\ (0:\ ),\ 0.0220 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5264088835 + 0.5149363573i\)
\(L(\frac12)\) \(\approx\) \(0.5264088835 + 0.5149363573i\)
\(L(1)\) \(\approx\) \(0.5997567270 + 0.2896864450i\)
\(L(1)\) \(\approx\) \(0.5997567270 + 0.2896864450i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
47 \( 1 \)
good2 \( 1 + (-0.460 + 0.887i)T \)
5 \( 1 + (-0.917 + 0.398i)T \)
11 \( 1 + (-0.962 - 0.269i)T \)
13 \( 1 + (-0.203 + 0.979i)T \)
17 \( 1 + (0.962 - 0.269i)T \)
19 \( 1 + (0.917 + 0.398i)T \)
23 \( 1 + (-0.460 - 0.887i)T \)
29 \( 1 + (-0.203 - 0.979i)T \)
31 \( 1 + (0.334 - 0.942i)T \)
37 \( 1 + (-0.0682 + 0.997i)T \)
41 \( 1 + (-0.990 - 0.136i)T \)
43 \( 1 + (-0.576 - 0.816i)T \)
53 \( 1 + (0.990 + 0.136i)T \)
59 \( 1 + (-0.576 + 0.816i)T \)
61 \( 1 + (0.0682 + 0.997i)T \)
67 \( 1 + (0.854 + 0.519i)T \)
71 \( 1 + (-0.460 - 0.887i)T \)
73 \( 1 + (0.775 + 0.631i)T \)
79 \( 1 + (0.682 + 0.730i)T \)
83 \( 1 + (0.962 + 0.269i)T \)
89 \( 1 + (-0.917 + 0.398i)T \)
97 \( 1 + (0.334 + 0.942i)T \)
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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

−21.34806044998134678368021916173, −20.48765390439357983843847467660, −19.966008679669434313617497092656, −19.3492966912023338700481188363, −18.35123803901438127058703456621, −17.87303311396096626332847405854, −16.86615603980218538171595416976, −16.035563924633380109887857495218, −15.39628791910964470074052672877, −14.24163097502232840996630650939, −13.18468459979793466247257594554, −12.55256382887134514099648481679, −11.92846182242701918037260576701, −11.0370928602610893863060427537, −10.27178377444423092153520655021, −9.504935853596206357478204176756, −8.42830015589682779093738869990, −7.81581268339234658458284250160, −7.21695354984769315153726964140, −5.367440159593011603279946725522, −4.85254956563455109915376818263, −3.48418842265767190099015937759, −3.12131178766917654371100126812, −1.70417018836588913026990467281, −0.57684220072865735471553743242, 0.77545970156956942677320980172, 2.319297273002185192695505179992, 3.591534112702947991781706545629, 4.54510682780257618809239055338, 5.45531490533210274087319536891, 6.42923315309520824914387764012, 7.331001413700706078904261681597, 7.91896639850860841784656778895, 8.6285055394101702437942369362, 9.82660992424812037134133436665, 10.34980517367362099553027260650, 11.482930852297983801974734736472, 12.14268695721860930516762930130, 13.493500032988532530952329517, 14.08368099592045214091652324281, 15.02007860458708832203270594992, 15.57063569562148114835229685440, 16.48545600683446311559807227080, 16.81956797505064560048613651157, 18.24926951994292903340673915083, 18.612084849303053689633372828470, 19.1752394298552204741860618736, 20.190057437769375180698809207000, 21.00253103405846497449834523437, 22.24389320829223352246487361835

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