↑

Properties

Label 1-287-287.209-r1-0-0
Degree $1$
Conductor $287$
Sign $0.244 - 0.969i$
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  + (0.309 − 0.951i)2-s + 3-s + (−0.809 − 0.587i)4-s + (0.809 + 0.587i)5-s + (0.309 − 0.951i)6-s + (−0.809 + 0.587i)8-s + 9-s + (0.809 − 0.587i)10-s + (0.809 − 0.587i)11-s + (−0.809 − 0.587i)12-s + (0.309 − 0.951i)13-s + (0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + (−0.809 + 0.587i)17-s + (0.309 − 0.951i)18-s + (0.309 + 0.951i)19-s + ⋯
L(s)  = 1  + (0.309 − 0.951i)2-s + 3-s + (−0.809 − 0.587i)4-s + (0.809 + 0.587i)5-s + (0.309 − 0.951i)6-s + (−0.809 + 0.587i)8-s + 9-s + (0.809 − 0.587i)10-s + (0.809 − 0.587i)11-s + (−0.809 − 0.587i)12-s + (0.309 − 0.951i)13-s + (0.809 + 0.587i)15-s + (0.309 + 0.951i)16-s + (−0.809 + 0.587i)17-s + (0.309 − 0.951i)18-s + (0.309 + 0.951i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.244 - 0.969i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (209, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.244 - 0.969i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.003045009 - 2.339025632i\)
\(L(\frac12)\) \(\approx\) \(3.003045009 - 2.339025632i\)
\(L(1)\) \(\approx\) \(1.770618733 - 0.9090010584i\)
\(L(1)\) \(\approx\) \(1.770618733 - 0.9090010584i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.309 - 0.951i)T \)
3 \( 1 + T \)
5 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 + (0.809 - 0.587i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
17 \( 1 + (-0.809 + 0.587i)T \)
19 \( 1 + (0.309 + 0.951i)T \)
23 \( 1 + (0.309 - 0.951i)T \)
29 \( 1 + (0.809 + 0.587i)T \)
31 \( 1 + (0.809 - 0.587i)T \)
37 \( 1 + (-0.809 - 0.587i)T \)
43 \( 1 + (0.309 - 0.951i)T \)
47 \( 1 + (0.309 - 0.951i)T \)
53 \( 1 + (0.809 + 0.587i)T \)
59 \( 1 + (-0.309 + 0.951i)T \)
61 \( 1 + (-0.309 - 0.951i)T \)
67 \( 1 + (0.809 + 0.587i)T \)
71 \( 1 + (0.809 - 0.587i)T \)
73 \( 1 - T \)
79 \( 1 - T \)
83 \( 1 - T \)
89 \( 1 + (0.309 + 0.951i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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

−25.46289707978671088684317574515, −24.61823523447180907316135015954, −24.15413603083413614749978253775, −22.87957941182333757039599180354, −21.74563259575121903014496233008, −21.202191897942122857488718468557, −20.13765239254513785611531405266, −19.13706871492958518587552157268, −17.87779474452016706003317470148, −17.29299393339595375950842644149, −16.067349889786262048637891010942, −15.40630260882975556250371145479, −14.17831208670253584318575480919, −13.751023454135928450563443383945, −12.92893159153452499780405384833, −11.771992281228108911887048111947, −9.77159821555027116096478856269, −9.18536168585267286743655679046, −8.48928270236073881904765589423, −7.12192858220444943888541998139, −6.45086609926545037725186264247, −4.900995139600237412238957820803, −4.212354192137728403547856753, −2.74495926322242743381698632632, −1.3155064234125187580719859875, 1.12701455412012126698988848339, 2.25971975992025947321823559452, 3.19262381385876653011475083190, 4.10131012893973455360586558697, 5.620031257100099499610330633049, 6.68899861298062752109801722532, 8.35171771716667356031405459575, 9.0685568956308770892630633034, 10.21374565483582184480481579683, 10.714735410793518213398176303631, 12.18382543469569385345230757230, 13.1710103182130617948547990728, 13.925278634760692413191754947959, 14.56455753193576539323599666085, 15.48925219282679119919551216077, 17.15305674654930422213056554998, 18.21273320219123209984949549859, 18.87985703419297513241058975158, 19.80489075956677933210225530311, 20.58935041393847013250502835186, 21.4264807529585988735209056098, 22.162598846669515462829421729095, 22.96897100488907668978439281261, 24.443015006958818986320866738379, 24.99832084584431804689024908155

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