Properties

Label 1-287-287.235-r1-0-0
Degree $1$
Conductor $287$
Sign $0.670 + 0.742i$
Analytic cond. $30.8424$
Root an. cond. $30.8424$
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  + (−0.207 + 0.978i)2-s + (−0.258 + 0.965i)3-s + (−0.913 − 0.406i)4-s + (0.994 − 0.104i)5-s + (−0.891 − 0.453i)6-s + (0.587 − 0.809i)8-s + (−0.866 − 0.5i)9-s + (−0.104 + 0.994i)10-s + (−0.777 + 0.629i)11-s + (0.629 − 0.777i)12-s + (0.453 − 0.891i)13-s + (−0.156 + 0.987i)15-s + (0.669 + 0.743i)16-s + (−0.629 − 0.777i)17-s + (0.669 − 0.743i)18-s + (−0.998 − 0.0523i)19-s + ⋯
L(s)  = 1  + (−0.207 + 0.978i)2-s + (−0.258 + 0.965i)3-s + (−0.913 − 0.406i)4-s + (0.994 − 0.104i)5-s + (−0.891 − 0.453i)6-s + (0.587 − 0.809i)8-s + (−0.866 − 0.5i)9-s + (−0.104 + 0.994i)10-s + (−0.777 + 0.629i)11-s + (0.629 − 0.777i)12-s + (0.453 − 0.891i)13-s + (−0.156 + 0.987i)15-s + (0.669 + 0.743i)16-s + (−0.629 − 0.777i)17-s + (0.669 − 0.743i)18-s + (−0.998 − 0.0523i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.670 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.670 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.670 + 0.742i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.670 + 0.742i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.260818985 + 0.5603110382i\)
\(L(\frac12)\) \(\approx\) \(1.260818985 + 0.5603110382i\)
\(L(1)\) \(\approx\) \(0.7841562945 + 0.4845252386i\)
\(L(1)\) \(\approx\) \(0.7841562945 + 0.4845252386i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.207 + 0.978i)T \)
3 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (0.994 - 0.104i)T \)
11 \( 1 + (-0.777 + 0.629i)T \)
13 \( 1 + (0.453 - 0.891i)T \)
17 \( 1 + (-0.629 - 0.777i)T \)
19 \( 1 + (-0.998 - 0.0523i)T \)
23 \( 1 + (0.978 + 0.207i)T \)
29 \( 1 + (0.987 + 0.156i)T \)
31 \( 1 + (0.104 - 0.994i)T \)
37 \( 1 + (-0.104 - 0.994i)T \)
43 \( 1 + (0.951 - 0.309i)T \)
47 \( 1 + (0.838 - 0.544i)T \)
53 \( 1 + (0.358 + 0.933i)T \)
59 \( 1 + (0.669 - 0.743i)T \)
61 \( 1 + (0.743 - 0.669i)T \)
67 \( 1 + (-0.933 + 0.358i)T \)
71 \( 1 + (0.156 + 0.987i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.965 + 0.258i)T \)
83 \( 1 + T \)
89 \( 1 + (0.0523 - 0.998i)T \)
97 \( 1 + (-0.156 + 0.987i)T \)
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

−25.38892307547433684054633913747, −24.08264723826966817546185659460, −23.38973125403112863305202145422, −22.31804612161684885570272791170, −21.4004664879524080559597678865, −20.839372925418288223960030573384, −19.396149789357455640186536217194, −18.9398890291363285101497751473, −17.98306728532180873791134713656, −17.36148834386686333795274424813, −16.44369205789690576636258984398, −14.56330500226023119903296150492, −13.55198757550222771925140459655, −13.15403509954260165433154514999, −12.15827492053283874827045467491, −10.96156494639871029397858452040, −10.46378393569104715754074305677, −8.93850193719863141771076100133, −8.36030924817895824960698715527, −6.82076232568274296057747142751, −5.85810975036572328689085696676, −4.638817075712597985324389930720, −2.93319856969289033221035965103, −2.04651307698906800971753007693, −1.020185020796230624419565289333, 0.566736522920310258804888981975, 2.60146622643599860791285883452, 4.27792883756624929297867180468, 5.20604606427395276711208321105, 5.9033468784450803710443606385, 7.05655199572053277546456969332, 8.47865019148125847619717617704, 9.27044703102883998306721185558, 10.19216212789748367557083078563, 10.85073137053993227505126618364, 12.7113338374337046712563789358, 13.53531175933816230023907537017, 14.623251781390478473818443132746, 15.44635105339218981132018133458, 16.12599574641984017337965763191, 17.35082379732703142474010156479, 17.60092555218245426955931979191, 18.686497925423550505615124111066, 20.20824867706335995061274936386, 21.03941771226529339287454368897, 21.95115830819965297432958414268, 22.842143920136660564418166524036, 23.46420374344984410891335383609, 24.85025355150813719556638076932, 25.48567473802533421291077978861

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