↑

Properties

Label 1-287-287.229-r0-0-0
Degree $1$
Conductor $287$
Sign $-0.806 + 0.591i$
Analytic cond. $1.33282$
Root an. cond. $1.33282$
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  + (0.743 + 0.669i)2-s + (−0.258 − 0.965i)3-s + (0.104 + 0.994i)4-s + (−0.406 + 0.913i)5-s + (0.453 − 0.891i)6-s + (−0.587 + 0.809i)8-s + (−0.866 + 0.5i)9-s + (−0.913 + 0.406i)10-s + (−0.358 + 0.933i)11-s + (0.933 − 0.358i)12-s + (−0.891 − 0.453i)13-s + (0.987 + 0.156i)15-s + (−0.978 + 0.207i)16-s + (−0.933 − 0.358i)17-s + (−0.978 − 0.207i)18-s + (0.838 + 0.544i)19-s + ⋯
L(s)  = 1  + (0.743 + 0.669i)2-s + (−0.258 − 0.965i)3-s + (0.104 + 0.994i)4-s + (−0.406 + 0.913i)5-s + (0.453 − 0.891i)6-s + (−0.587 + 0.809i)8-s + (−0.866 + 0.5i)9-s + (−0.913 + 0.406i)10-s + (−0.358 + 0.933i)11-s + (0.933 − 0.358i)12-s + (−0.891 − 0.453i)13-s + (0.987 + 0.156i)15-s + (−0.978 + 0.207i)16-s + (−0.933 − 0.358i)17-s + (−0.978 − 0.207i)18-s + (0.838 + 0.544i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.806 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.806 + 0.591i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ -0.806 + 0.591i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3070522568 + 0.9375648927i\)
\(L(\frac12)\) \(\approx\) \(0.3070522568 + 0.9375648927i\)
\(L(1)\) \(\approx\) \(0.9115158966 + 0.5251954113i\)
\(L(1)\) \(\approx\) \(0.9115158966 + 0.5251954113i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.743 + 0.669i)T \)
3 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (-0.406 + 0.913i)T \)
11 \( 1 + (-0.358 + 0.933i)T \)
13 \( 1 + (-0.891 - 0.453i)T \)
17 \( 1 + (-0.933 - 0.358i)T \)
19 \( 1 + (0.838 + 0.544i)T \)
23 \( 1 + (-0.669 + 0.743i)T \)
29 \( 1 + (-0.156 + 0.987i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (-0.951 + 0.309i)T \)
47 \( 1 + (-0.998 + 0.0523i)T \)
53 \( 1 + (0.777 + 0.629i)T \)
59 \( 1 + (0.978 + 0.207i)T \)
61 \( 1 + (0.207 + 0.978i)T \)
67 \( 1 + (0.629 - 0.777i)T \)
71 \( 1 + (-0.987 + 0.156i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (0.965 + 0.258i)T \)
83 \( 1 - T \)
89 \( 1 + (0.544 - 0.838i)T \)
97 \( 1 + (-0.987 - 0.156i)T \)
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

−24.69242053900673935328649774913, −24.15563962266356803846454198041, −23.24571720884755718668414820943, −22.20316252904793228929387438851, −21.609100993314212890002224884070, −20.76994525265255116188956231845, −19.96266260887022856716397526835, −19.27775565812350310947887098667, −17.818903260191998192844546035604, −16.58987254260125803253490083592, −15.8891304257558629050444648121, −15.06611641845182207191476349544, −13.963678993474925221861104174656, −13.02152998247167366138516489911, −11.83336769783271927198189653481, −11.363650520806865562361013863168, −10.17602668582035548157245707635, −9.318254246108401845676649849, −8.29849522370619673420613858476, −6.44343128820811298821987724173, −5.29376843827665224563842279563, −4.59867815288379721224045402247, −3.70414957461724454541901807711, −2.423644036497895732892889023663, −0.499382776677964936547462608730, 2.206356435918452650698212743, 3.13110697753629485324627977024, 4.62855112954976618378031419214, 5.72153448475676486020105871337, 6.84266383381104600451121584758, 7.40826650641323050953415677187, 8.18185501876672424767225479134, 9.929110123967931924445534645978, 11.37229707944163289534811043725, 12.01811278497226994261946948187, 12.96908394517370709737111720111, 13.88118616309457105830478453484, 14.77850524377687654022761800942, 15.546334226815865651655979694013, 16.70888771987945593437294039805, 17.971597605746848714626480886639, 18.06948778478651176364638357618, 19.61774442592054179433865873140, 20.33709468181655363969270002674, 21.90584706556103527843590927166, 22.553747913971011605687864319243, 23.15893313599505860652165830185, 24.03640978498038438880626223514, 24.84054906481553833040256997990, 25.66872674341499405260071528065

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