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Properties

Label 1-287-287.114-r0-0-0
Degree $1$
Conductor $287$
Sign $-0.698 + 0.715i$
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  + (0.5 + 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + i·6-s − 8-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.866 + 0.5i)12-s − i·13-s + i·15-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.5 + 0.866i)18-s + (0.866 − 0.5i)19-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + i·6-s − 8-s + (0.5 + 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.866 + 0.5i)11-s + (−0.866 + 0.5i)12-s − i·13-s + i·15-s + (−0.5 − 0.866i)16-s + (−0.866 − 0.5i)17-s + (−0.5 + 0.866i)18-s + (0.866 − 0.5i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.698 + 0.715i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.698 + 0.715i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (114, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ -0.698 + 0.715i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8410654292 + 1.995997825i\)
\(L(\frac12)\) \(\approx\) \(0.8410654292 + 1.995997825i\)
\(L(1)\) \(\approx\) \(1.221640341 + 1.266446701i\)
\(L(1)\) \(\approx\) \(1.221640341 + 1.266446701i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
11 \( 1 + (0.866 + 0.5i)T \)
13 \( 1 - iT \)
17 \( 1 + (-0.866 - 0.5i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 - iT \)
31 \( 1 + (-0.5 + 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
43 \( 1 - T \)
47 \( 1 + (0.866 - 0.5i)T \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + (-0.5 + 0.866i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 - iT \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.866 + 0.5i)T \)
97 \( 1 + iT \)
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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

−24.920285320488689912881957955048, −24.188178298387768637293166827771, −23.68539507911867624529750982071, −22.08927733512990389208624527931, −21.54725751456606528187987849543, −20.475459748516668134620512293408, −19.97316600827117427449484270375, −19.10104462270368332722883182506, −18.20998210329458880554426460193, −17.11599693340474910446353375310, −15.81806860395485542333531108228, −14.548592381854818837674428480074, −13.842475707115460972789661077321, −13.216105351415036973680618904585, −12.212177164020247235756350101845, −11.45020118580329824848005647014, −9.85620339546545518899717064122, −9.16608842068966389860019264324, −8.42746183640546846356871606262, −6.75572148526960171198447421690, −5.69737275431326037213655335353, −4.32317632956241314158956200137, −3.46372246848274278548501394462, −1.97620786782047186515269281565, −1.32022438192436761390778333387, 2.34523774677468680222202537625, 3.30528281302135132433130577661, 4.36042385644151902348485618472, 5.527016168947692222538807947085, 6.78598975917211425140334014985, 7.52248398802947000757808977322, 8.75637821312475959582042255237, 9.58463660583532524471599022721, 10.65215055045963079227097171120, 12.07957754819957709483299626201, 13.41289260322917879641010231932, 13.94369259367799464074630142832, 14.90578375555976395793883542863, 15.395002578357692772581215671811, 16.46239748310014461627410571059, 17.673373360460533902593656163323, 18.27305389853710203040723283844, 19.691766603416626065216233901969, 20.54363540471217443430260415670, 21.64342918837061947251401025182, 22.36017756553099968513842850421, 22.84364321292061373853204822281, 24.48148586904737180279458763334, 25.0064672829440243177432040603, 25.72632397885330153212997927388

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