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Properties

Label 1-1287-1287.799-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.881 - 0.472i$
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  + (0.207 − 0.978i)2-s + (−0.913 − 0.406i)4-s + (0.743 − 0.669i)5-s + (0.587 + 0.809i)7-s + (−0.587 + 0.809i)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 + (0.406 + 0.913i)19-s + (−0.951 + 0.309i)20-s − 23-s + (0.104 − 0.994i)25-s + (−0.207 − 0.978i)28-s + (0.104 + 0.994i)29-s + ⋯
L(s)  = 1  + (0.207 − 0.978i)2-s + (−0.913 − 0.406i)4-s + (0.743 − 0.669i)5-s + (0.587 + 0.809i)7-s + (−0.587 + 0.809i)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 + (0.406 + 0.913i)19-s + (−0.951 + 0.309i)20-s − 23-s + (0.104 − 0.994i)25-s + (−0.207 − 0.978i)28-s + (0.104 + 0.994i)29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.881 - 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.881 - 0.472i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (799, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.881 - 0.472i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.759113796 - 0.4417998257i\)
\(L(\frac12)\) \(\approx\) \(1.759113796 - 0.4417998257i\)
\(L(1)\) \(\approx\) \(1.192896889 - 0.4714656735i\)
\(L(1)\) \(\approx\) \(1.192896889 - 0.4714656735i\)

Euler product

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

−21.23039903346829835255873430243, −20.51881415530520154155427055338, −19.38013227973597680546719254049, −18.46068331943587247422436653654, −17.78414585870600193597334016983, −17.36796728932221152623723466284, −16.47592884881595242719924496793, −15.73375522812320098024894587025, −14.74933902048925900998646658877, −14.19050684468569747600112059398, −13.66371355332827270068321740164, −12.94000666039208990009891595760, −11.730630488406763488106752091473, −10.91807653695502849663779966761, −9.81075140882035268765447729243, −9.45053402058551769730759156932, −8.11665142529057885043388600662, −7.51714919082684002306355144108, −6.781736210889006616491967741562, −5.93163323418520013643809715776, −5.14348438085075587325733266494, −4.24282377266724344084439695379, −3.30293483364110055559671373001, −2.17248427839427717964797437508, −0.72079613665578499475505008227, 1.29298464813829842352967624006, 1.77805181023896282406080924088, 2.811124781292579030013573235520, 3.8572943707549825222517287806, 4.86507498714834338706772606663, 5.548536882277392231046201540865, 6.137151003784106814334850326813, 7.83637453146294609255022256844, 8.6072681369905652385225861226, 9.2195473939214720830628307867, 10.13929908392403534686752516339, 10.70185551055015060799296054561, 11.970936140320103660130702270327, 12.25501597165543683670227136737, 13.0069933699423107796161120319, 14.15724806903788366101396901664, 14.31917585804285055428845318118, 15.484059707812797063869162642045, 16.505828571732438129814718824222, 17.36578882461416325982920315208, 18.11180767893074625267744288574, 18.57776505018159255756341943212, 19.605976329915092296387254333237, 20.29195808159906492810669681180, 21.15101984939736753160542318568

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