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Properties

Label 1-287-287.94-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

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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: $\chi_{287} (94, \cdot )$
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 \( 1 + (-0.987 + 0.156i)T \)
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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

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

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