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Properties

Label 1-1287-1287.1174-r1-0-0
Degree $1$
Conductor $1287$
Sign $0.548 + 0.835i$
Analytic cond. $138.307$
Root an. cond. $138.307$
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  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (−0.669 + 0.743i)5-s + (−0.104 + 0.994i)7-s + (−0.809 + 0.587i)8-s + (0.5 + 0.866i)10-s + (0.913 + 0.406i)14-s + (0.309 + 0.951i)16-s + (0.978 + 0.207i)17-s + (−0.104 − 0.994i)19-s + (0.978 − 0.207i)20-s + (−0.5 + 0.866i)23-s + (−0.104 − 0.994i)25-s + (0.669 − 0.743i)28-s + (0.809 + 0.587i)29-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + (−0.809 − 0.587i)4-s + (−0.669 + 0.743i)5-s + (−0.104 + 0.994i)7-s + (−0.809 + 0.587i)8-s + (0.5 + 0.866i)10-s + (0.913 + 0.406i)14-s + (0.309 + 0.951i)16-s + (0.978 + 0.207i)17-s + (−0.104 − 0.994i)19-s + (0.978 − 0.207i)20-s + (−0.5 + 0.866i)23-s + (−0.104 − 0.994i)25-s + (0.669 − 0.743i)28-s + (0.809 + 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.548 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.548 + 0.835i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.548 + 0.835i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ 0.548 + 0.835i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.207224006 + 0.6516373983i\)
\(L(\frac12)\) \(\approx\) \(1.207224006 + 0.6516373983i\)
\(L(1)\) \(\approx\) \(0.9571888377 - 0.1823345251i\)
\(L(1)\) \(\approx\) \(0.9571888377 - 0.1823345251i\)

Euler product

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

−20.72118358251074252445354729673, −20.06840221109261871483463932328, −19.0459787495493432419446861716, −18.39540239158479471305644764901, −17.14366748557923166016504134748, −16.891750950497055184310811615207, −16.11815121097864009329165604731, −15.51151963762121634767544683309, −14.47865474605880586598062009022, −13.91939119315390458924024455156, −13.07730153390554598950746778634, −12.29556432235767360511488644165, −11.70866994086554973739796208983, −10.28711905807025682259826260582, −9.70680897460321035053437939294, −8.391454160562728933829577818758, −8.10972713409972691974444145268, −7.22668980734050202360480423165, −6.38949011017922031908760450846, −5.41943988289936335132567230875, −4.486046850364389918264272422760, −3.97869439607182722588754764351, −3.02972330961441570578005292030, −1.17642148186280895656744901343, −0.32877606241045961437003739573, 0.90512883750245538421499295081, 2.2179815435511355495488462450, 2.91719205453535979762147717796, 3.667369689437231316914035948258, 4.659375152897948202754852665934, 5.6054645002952800996221501893, 6.39288816332770300047011319496, 7.57741223716144714672439051842, 8.47942126928679020191467894263, 9.31136175555552107547907576042, 10.15216618267931022448281494188, 10.94048528119732147549747590589, 11.733973492149568621470492145, 12.18166909301246493767757645217, 13.03588627982584006428539396062, 14.05916520112176216645655698408, 14.66367788453377297586834661979, 15.44775880840507664499616389700, 16.05716798906193341474734044471, 17.56314234857635892814242963388, 18.04600359145566496513918127201, 18.97786559519346002489869262674, 19.3351102841210324218275348939, 20.00684032999694270107706843626, 21.23579310802301884059949695455

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