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Properties

Label 1-1287-1287.347-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.827 - 0.560i$
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  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (0.104 − 0.994i)5-s + (0.978 − 0.207i)7-s + (0.309 + 0.951i)8-s + (0.5 + 0.866i)10-s + (−0.669 + 0.743i)14-s + (−0.809 − 0.587i)16-s + (0.913 − 0.406i)17-s + (0.978 + 0.207i)19-s + (−0.913 − 0.406i)20-s + (0.5 − 0.866i)23-s + (−0.978 − 0.207i)25-s + (0.104 − 0.994i)28-s + (0.309 − 0.951i)29-s + ⋯
L(s)  = 1  + (−0.809 + 0.587i)2-s + (0.309 − 0.951i)4-s + (0.104 − 0.994i)5-s + (0.978 − 0.207i)7-s + (0.309 + 0.951i)8-s + (0.5 + 0.866i)10-s + (−0.669 + 0.743i)14-s + (−0.809 − 0.587i)16-s + (0.913 − 0.406i)17-s + (0.978 + 0.207i)19-s + (−0.913 − 0.406i)20-s + (0.5 − 0.866i)23-s + (−0.978 − 0.207i)25-s + (0.104 − 0.994i)28-s + (0.309 − 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.827 - 0.560i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (347, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.827 - 0.560i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.245657226 - 0.3821159780i\)
\(L(\frac12)\) \(\approx\) \(1.245657226 - 0.3821159780i\)
\(L(1)\) \(\approx\) \(0.9191722279 - 0.05230247284i\)
\(L(1)\) \(\approx\) \(0.9191722279 - 0.05230247284i\)

Euler product

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

−21.01883569025127092785174643551, −20.34379605287610183198765076125, −19.3386405531620246547153657770, −18.80152343269842314795432914432, −18.07100716843972801059912791821, −17.53447006948301247263152800215, −16.80640907442375509736819285292, −15.68428322101758487790742732928, −15.06333983893102111772308823248, −14.08336209759505930131520087404, −13.437771139780437998157234467874, −12.02619500131708098508938151793, −11.81365361373038483283064903741, −10.74063478919592100983315658663, −10.36717560132097000823041195953, −9.36867130547598763458066538421, −8.55446541230717057699673769936, −7.57449477750787528269345821187, −7.2067775469370273705249730886, −5.98564734262885873950516590270, −4.96181042534755618615471848073, −3.6665143624002996791513509074, −3.01275587040939538528664265137, −2.00047579850663560860572179875, −1.15063343213817494566008726504, 0.84182288778403461052342656310, 1.437587024218030270304766471713, 2.64384112793246776228957789176, 4.2356479541280857021796447284, 5.09955926116420091054314200106, 5.59942464425588718166350256712, 6.77892447260071857484624469385, 7.7167599044828561539820939432, 8.266522685479932740233139756529, 8.99132810798861721464733238896, 9.87407686172785683630814521488, 10.54258255392537165975769016870, 11.67936985886780154464213778315, 12.12575317558324154715590681379, 13.51516676119493629453000340433, 14.08274532234462369609855618025, 14.90659506888944337926591199202, 15.81641873000590393397024481213, 16.4293012094388572365478222026, 17.21039632895649260724792997220, 17.65758757420859342987163104774, 18.574307456253321550693456694752, 19.25763497391951766014182856539, 20.32139307527766056776128785798, 20.658947768601146034442913546

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