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Properties

Label 1-1287-1287.644-r1-0-0
Degree $1$
Conductor $1287$
Sign $-0.296 - 0.954i$
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  + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s + (0.994 + 0.104i)5-s + (−0.207 − 0.978i)7-s + (0.951 − 0.309i)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 + (−0.207 + 0.978i)19-s + (−0.406 + 0.913i)20-s + (−0.5 + 0.866i)23-s + (0.978 + 0.207i)25-s + (0.994 + 0.104i)28-s + (0.309 − 0.951i)29-s + ⋯
L(s)  = 1  + (−0.587 − 0.809i)2-s + (−0.309 + 0.951i)4-s + (0.994 + 0.104i)5-s + (−0.207 − 0.978i)7-s + (0.951 − 0.309i)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 + (−0.207 + 0.978i)19-s + (−0.406 + 0.913i)20-s + (−0.5 + 0.866i)23-s + (0.978 + 0.207i)25-s + (0.994 + 0.104i)28-s + (0.309 − 0.951i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.296 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.296 - 0.954i$
Analytic conductor: \(138.307\)
Root analytic conductor: \(138.307\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (644, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (1:\ ),\ -0.296 - 0.954i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8481366242 - 1.151702277i\)
\(L(\frac12)\) \(\approx\) \(0.8481366242 - 1.151702277i\)
\(L(1)\) \(\approx\) \(0.7995184645 - 0.3490000074i\)
\(L(1)\) \(\approx\) \(0.7995184645 - 0.3490000074i\)

Euler product

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

−21.07731958067757474555199636720, −20.09312531578724661879952435, −19.41560143116297667124076415652, −18.30055408018789974717721801075, −18.11449263136103317356519615129, −17.33226721929572500146244543972, −16.37720094599056563271982347305, −15.89098348791708522753189401824, −14.884883520840311954194663632510, −14.410158669741584183463178300361, −13.32941906913047721911577684545, −12.86687657938822144593104534784, −11.56810249869978054496044954763, −10.69366207314240900262929769933, −9.84467812836602527921520858701, −8.9908901717916074394365642382, −8.795231892084402639249826906486, −7.55278384008106325009075594379, −6.53600285473746007136202192956, −6.08443159063850268672837690661, −5.181722409776390493705003888453, −4.45887412077958620637177880940, −2.6743049017986893036660481533, −2.04962470377076903623953571288, −0.79587022124832652567430783272, 0.42856212714529861222999813245, 1.56474759881968264871239453865, 2.19513988488713486006907987438, 3.41547398688340350341021184709, 4.09141915080031110849563494155, 5.21414001623923875191019955352, 6.38764375852973144358177537278, 7.13269700732851743274678372795, 8.12967918544338942008742081996, 8.93587981510525837365716008832, 9.90933754171714187663974425429, 10.26585156654077369589867282748, 11.014801418529549981215323829413, 11.989561807089174469994784376875, 12.874649634881286704583516434981, 13.61476677177612927517774132201, 13.99186438451890305022032881167, 15.26520419338831771883625968090, 16.40019784041448440693939399755, 16.99530390375984711191267817859, 17.619322938118643379712220382744, 18.22951302422152235697587227375, 19.16923275519249319982689513291, 19.84930551322579876016934580136, 20.56755596282322441759255062355

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