Properties

Label 1-1287-1287.140-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.0828 - 0.996i$
Analytic cond. $5.97680$
Root an. cond. $5.97680$
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  + (0.978 − 0.207i)2-s + (0.913 − 0.406i)4-s + (0.669 − 0.743i)5-s + (−0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (0.5 − 0.866i)10-s + (−0.913 − 0.406i)14-s + (0.669 − 0.743i)16-s + (0.669 − 0.743i)17-s + (0.913 + 0.406i)19-s + (0.309 − 0.951i)20-s − 23-s + (−0.104 − 0.994i)25-s + (−0.978 − 0.207i)28-s + (−0.104 + 0.994i)29-s + ⋯
L(s)  = 1  + (0.978 − 0.207i)2-s + (0.913 − 0.406i)4-s + (0.669 − 0.743i)5-s + (−0.809 − 0.587i)7-s + (0.809 − 0.587i)8-s + (0.5 − 0.866i)10-s + (−0.913 − 0.406i)14-s + (0.669 − 0.743i)16-s + (0.669 − 0.743i)17-s + (0.913 + 0.406i)19-s + (0.309 − 0.951i)20-s − 23-s + (−0.104 − 0.994i)25-s + (−0.978 − 0.207i)28-s + (−0.104 + 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0828 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.0828 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.0828 - 0.996i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (140, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.0828 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.122537358 - 2.306252442i\)
\(L(\frac12)\) \(\approx\) \(2.122537358 - 2.306252442i\)
\(L(1)\) \(\approx\) \(1.843234616 - 0.8589779582i\)
\(L(1)\) \(\approx\) \(1.843234616 - 0.8589779582i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (0.978 - 0.207i)T \)
5 \( 1 + (0.669 - 0.743i)T \)
7 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (0.669 - 0.743i)T \)
19 \( 1 + (0.913 + 0.406i)T \)
23 \( 1 - T \)
29 \( 1 + (-0.104 + 0.994i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (-0.913 + 0.406i)T \)
41 \( 1 + (0.809 - 0.587i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.104 - 0.994i)T \)
53 \( 1 + (-0.309 + 0.951i)T \)
59 \( 1 + (0.913 - 0.406i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 - T \)
71 \( 1 + (0.669 - 0.743i)T \)
73 \( 1 + (-0.809 - 0.587i)T \)
79 \( 1 + (-0.669 - 0.743i)T \)
83 \( 1 + (0.978 + 0.207i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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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

−21.38338794315801955787650050406, −20.78423192935091630446409414973, −19.61610580782920189921441488776, −19.13164369220050387587608058375, −18.066016405294277881912663670225, −17.36277952818317520689605213302, −16.383348511828564813487719123211, −15.71785487124217769193498452033, −14.99809087731930646335969969690, −14.23129588333936394721157162223, −13.56957933085807457809369371007, −12.85252564557705098726355656192, −12.01256770503429648403231382109, −11.33330354960475776749256535720, −10.197742726609191112211958590547, −9.764141845711394248182830892567, −8.46994555994660605949334329299, −7.48545270693619515264256025387, −6.62464896041664247265733968897, −5.95679258100598905354879981059, −5.4278596878113309277286988766, −4.15868612276309460387243244242, −3.20147501228867712495151822558, −2.630881233473420017121558690995, −1.61526832803574478653429740817, 0.866447045794103701313905771882, 1.821696554940092484254115302354, 2.97831168970967590533415077332, 3.71261136804641209903379675387, 4.74584130212842940479594938502, 5.46439930395852100022649846011, 6.2418099687887146644963668059, 7.0787165169794336435456951447, 8.00013500314504326872965538659, 9.315117060430422032270066201410, 9.97604397014421657627730644848, 10.57283246092654015357696312791, 11.91095098711573322298627310755, 12.26667128272779526982234339233, 13.22600849663871318767522199841, 13.80235138171501869428156479605, 14.27194453486155300472951750449, 15.509382507847635404208212129306, 16.299155768812298924238496470771, 16.59398247711616714125439085524, 17.67316307651939604042004831095, 18.682590573132104662394200206664, 19.57105084296772750553236300019, 20.43226435260383265984205786293, 20.6027333173237019433518533798

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