# 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 $0$

# Related objects

## 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$$
show less
$$L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−24.920285320488689912881957955048, −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