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Properties

Label 1-1287-1287.106-r0-0-0
Degree $1$
Conductor $1287$
Sign $-0.537 + 0.843i$
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  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (0.743 − 0.669i)5-s + (−0.994 + 0.104i)7-s + (−0.587 + 0.809i)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 + (−0.994 − 0.104i)19-s + (0.207 − 0.978i)20-s + (0.5 + 0.866i)23-s + (0.104 − 0.994i)25-s + (−0.743 + 0.669i)28-s + (0.809 − 0.587i)29-s + ⋯
L(s)  = 1  + (−0.951 + 0.309i)2-s + (0.809 − 0.587i)4-s + (0.743 − 0.669i)5-s + (−0.994 + 0.104i)7-s + (−0.587 + 0.809i)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 + (−0.994 − 0.104i)19-s + (0.207 − 0.978i)20-s + (0.5 + 0.866i)23-s + (0.104 − 0.994i)25-s + (−0.743 + 0.669i)28-s + (0.809 − 0.587i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $-0.537 + 0.843i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (106, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ -0.537 + 0.843i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1887403284 + 0.3443014491i\)
\(L(\frac12)\) \(\approx\) \(0.1887403284 + 0.3443014491i\)
\(L(1)\) \(\approx\) \(0.5928119632 + 0.06299207796i\)
\(L(1)\) \(\approx\) \(0.5928119632 + 0.06299207796i\)

Euler product

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

−20.656374487239500296469079920176, −19.78136040559432062231585403683, −19.21433380477638501687085059247, −18.365938628990976859196281084363, −17.952763601270023673054865179762, −16.821094711133567515877468441712, −16.60473299551379743449402763052, −15.389535657552092648548127438539, −14.87386780224730823139731821117, −13.56444024429976496674889222959, −13.02324399456000702869346897036, −12.14442102356514385143311326716, −11.07008994053808287658468228460, −10.50889145214611334300095206227, −9.817832161789943764735132808953, −9.09556951013744230238584964849, −8.33336950249120826495990215043, −7.07241560375390049011681665356, −6.61582744459554450562299315483, −5.94357233808514417152091051446, −4.42181570852196967991620721577, −3.23851132123438716069845635468, −2.58896358578307311684677630128, −1.72045243969115817024775079567, −0.21329853890671281350024179782, 1.164053649085166760060318814028, 2.14203734888840181944912375145, 3.03996471257983766273190585205, 4.48571270600772258662686956378, 5.48697443352789823979919838980, 6.38058988232518227673422374682, 6.77577848465611125496225583430, 8.07086180354310307651865747601, 8.81312928966015740008409097221, 9.42412102233816044068766397423, 10.098944359731741683402356972467, 10.888787143204952565062660906095, 11.90362259762212524844260399070, 12.861933296573168956602176085841, 13.431415962935423123623938516817, 14.49760964292255996598023723698, 15.48482055355218053583995504641, 15.9814632224899934547302245941, 16.903195090515481618693445285736, 17.30859637399740640298429868709, 18.136799616704958696356107019686, 18.94356815978394633104590699657, 19.86298214136985576508416078879, 20.04755924229216521466977545379, 21.43014189884202257831202390086

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