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Properties

Label 1-287-287.135-r1-0-0
Degree $1$
Conductor $287$
Sign $-0.539 + 0.842i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.207 + 0.978i)2-s + (−0.965 + 0.258i)3-s + (−0.913 + 0.406i)4-s + (−0.994 − 0.104i)5-s + (−0.453 − 0.891i)6-s + (−0.587 − 0.809i)8-s + (0.866 − 0.5i)9-s + (−0.104 − 0.994i)10-s + (−0.629 + 0.777i)11-s + (0.777 − 0.629i)12-s + (0.891 − 0.453i)13-s + (0.987 − 0.156i)15-s + (0.669 − 0.743i)16-s + (−0.777 − 0.629i)17-s + (0.669 + 0.743i)18-s + (−0.0523 − 0.998i)19-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)2-s + (−0.965 + 0.258i)3-s + (−0.913 + 0.406i)4-s + (−0.994 − 0.104i)5-s + (−0.453 − 0.891i)6-s + (−0.587 − 0.809i)8-s + (0.866 − 0.5i)9-s + (−0.104 − 0.994i)10-s + (−0.629 + 0.777i)11-s + (0.777 − 0.629i)12-s + (0.891 − 0.453i)13-s + (0.987 − 0.156i)15-s + (0.669 − 0.743i)16-s + (−0.777 − 0.629i)17-s + (0.669 + 0.743i)18-s + (−0.0523 − 0.998i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.539 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.539 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.539 + 0.842i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (135, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ -0.539 + 0.842i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3458660740 + 0.6322098458i\)
\(L(\frac12)\) \(\approx\) \(0.3458660740 + 0.6322098458i\)
\(L(1)\) \(\approx\) \(0.5344221003 + 0.3387149315i\)
\(L(1)\) \(\approx\) \(0.5344221003 + 0.3387149315i\)

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 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (-0.994 - 0.104i)T \)
11 \( 1 + (-0.629 + 0.777i)T \)
13 \( 1 + (0.891 - 0.453i)T \)
17 \( 1 + (-0.777 - 0.629i)T \)
19 \( 1 + (-0.0523 - 0.998i)T \)
23 \( 1 + (0.978 - 0.207i)T \)
29 \( 1 + (-0.156 - 0.987i)T \)
31 \( 1 + (0.104 + 0.994i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (-0.951 - 0.309i)T \)
47 \( 1 + (-0.544 + 0.838i)T \)
53 \( 1 + (-0.933 - 0.358i)T \)
59 \( 1 + (0.669 + 0.743i)T \)
61 \( 1 + (-0.743 - 0.669i)T \)
67 \( 1 + (0.358 - 0.933i)T \)
71 \( 1 + (-0.987 - 0.156i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (-0.258 + 0.965i)T \)
83 \( 1 + T \)
89 \( 1 + (0.998 - 0.0523i)T \)
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

−24.56235586017558583814153058701, −23.58018862846051720571295772144, −23.301836593025208899426572981910, −22.30330810563666681548405323828, −21.45334343428394167828000940808, −20.56379769824757213287061030135, −19.33085427029154569863182410645, −18.76084377893410319351569452948, −18.0507700840261953072094165019, −16.77295396553368446996191022577, −15.89717100639664025014660925186, −14.77270185192219612647050403573, −13.39627457942262615868689885287, −12.755439174366704409794852319008, −11.684096714364925392129064894433, −11.06426538387316714426005322916, −10.44988641598487036014820043674, −8.88197704520000019064978229792, −7.89602954552293361706672102937, −6.46287207494612348375956578374, −5.3952815316497150462989883054, −4.26912171810512077459806375266, −3.348883512605048517077994573311, −1.67785444194561242975071535579, −0.4143034296867407192205874664, 0.6824887721083607260579180264, 3.278840816923923018092757667, 4.59488821495938811416314686134, 5.01526230879988585387727215312, 6.46267025186312887327857418839, 7.18114279516680898904559576308, 8.26637471480007930121533502489, 9.380715602943567534757209534583, 10.68612533644156980171518096116, 11.63576977773198449648214495557, 12.69730835509836616019789054672, 13.41472335382239336163144021241, 15.11647645414372285373832148185, 15.5355694819281736933188239922, 16.209300104054014319999705439681, 17.3040198018533295255095985455, 18.02816413615900299089039537088, 18.88120701424072078008244015683, 20.33369623194583029251881750133, 21.31546959423528644645007829157, 22.54794416753363493130310461440, 22.95797668628837703676210382801, 23.69089668000180677900867917258, 24.448693191630250922194503108288, 25.584010259174433264716322847555

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