↑

Properties

Label 1-1287-1287.1154-r0-0-0
Degree $1$
Conductor $1287$
Sign $0.944 + 0.329i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s − 8-s + (0.5 − 0.866i)10-s + (0.5 − 0.866i)14-s + 16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)20-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 + 0.866i)28-s + 29-s + ⋯
L(s)  = 1  − 2-s + 4-s + (−0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s − 8-s + (0.5 − 0.866i)10-s + (0.5 − 0.866i)14-s + 16-s + (−0.5 − 0.866i)17-s + (−0.5 − 0.866i)19-s + (−0.5 + 0.866i)20-s + (0.5 + 0.866i)23-s + (−0.5 − 0.866i)25-s + (−0.5 + 0.866i)28-s + 29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1287\)    =    \(3^{2} \cdot 11 \cdot 13\)
Sign: $0.944 + 0.329i$
Analytic conductor: \(5.97680\)
Root analytic conductor: \(5.97680\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1287} (1154, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1287,\ (0:\ ),\ 0.944 + 0.329i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6923158218 + 0.1175020336i\)
\(L(\frac12)\) \(\approx\) \(0.6923158218 + 0.1175020336i\)
\(L(1)\) \(\approx\) \(0.5997385463 + 0.09942098024i\)
\(L(1)\) \(\approx\) \(0.5997385463 + 0.09942098024i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
11 \( 1 \)
13 \( 1 \)
good2 \( 1 - T \)
5 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (-0.5 + 0.866i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (0.5 + 0.866i)T \)
29 \( 1 + T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 - T \)
59 \( 1 + T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 + (-0.5 - 0.866i)T \)
73 \( 1 + T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + (0.5 + 0.866i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (0.5 - 0.866i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−20.74644357664280265617949916606, −20.057992872657469658360001033811, −19.37116776004735660507223527355, −18.97117450618911047782495791333, −17.67637893992827251634820478888, −17.20727772333495993369951782282, −16.39031862915886687074500522618, −16.00511633399262364253271471690, −15.06594044921889121281071436235, −14.14620190248631674279303288133, −12.86913024076165759722928930575, −12.528615846958962015022309503577, −11.49389752234172579224384099467, −10.62686985334329655489085345959, −10.057996132401369211871579354195, −9.07900499847172133458685535627, −8.35130656105663883438368021212, −7.771609295467849234876881038960, −6.723371564983123369455019237966, −6.11898509056091280236437262105, −4.73970946789083099451096782084, −3.90174760970217805743449820536, −2.87538606966906874378991574757, −1.56792882585989256953457492310, −0.72793336444218625020303405753, 0.60248760580650511492991376247, 2.295379025707367506145542911143, 2.7066627110338273689870222004, 3.722331131821382784354098722614, 5.12545613260487814520622562643, 6.27829177116897667751131362822, 6.78667347842472650242690766442, 7.63334452707679056882046779160, 8.49883692142262004476903914245, 9.3268290409134346282338833464, 9.93569755960938706689252893806, 11.04110230137553884432499217820, 11.435868772188085342150639801875, 12.25013436241842491156207176569, 13.23680751879442873915547599734, 14.39171979069736827983074538281, 15.330224828320061483770101405921, 15.60310839132814487527925547575, 16.39501848823060668918395398517, 17.515387449370727084727198475505, 18.05150308322242018167475583240, 18.79634358927468155149097622404, 19.41218315231072715478088081010, 19.884158544736980111597897267382, 21.0210252132771832397965965083

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