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Properties

Label 1-287-287.48-r0-0-0
Degree $1$
Conductor $287$
Sign $0.439 + 0.898i$
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  + (−0.587 + 0.809i)2-s + (0.707 + 0.707i)3-s + (−0.309 − 0.951i)4-s + (0.951 − 0.309i)5-s + (−0.987 + 0.156i)6-s + (0.951 + 0.309i)8-s + i·9-s + (−0.309 + 0.951i)10-s + (0.891 − 0.453i)11-s + (0.453 − 0.891i)12-s + (−0.156 − 0.987i)13-s + (0.891 + 0.453i)15-s + (−0.809 + 0.587i)16-s + (−0.453 − 0.891i)17-s + (−0.809 − 0.587i)18-s + (−0.156 + 0.987i)19-s + ⋯
L(s)  = 1  + (−0.587 + 0.809i)2-s + (0.707 + 0.707i)3-s + (−0.309 − 0.951i)4-s + (0.951 − 0.309i)5-s + (−0.987 + 0.156i)6-s + (0.951 + 0.309i)8-s + i·9-s + (−0.309 + 0.951i)10-s + (0.891 − 0.453i)11-s + (0.453 − 0.891i)12-s + (−0.156 − 0.987i)13-s + (0.891 + 0.453i)15-s + (−0.809 + 0.587i)16-s + (−0.453 − 0.891i)17-s + (−0.809 − 0.587i)18-s + (−0.156 + 0.987i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.439 + 0.898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.439 + 0.898i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (48, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.439 + 0.898i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.178249817 + 0.7355494689i\)
\(L(\frac12)\) \(\approx\) \(1.178249817 + 0.7355494689i\)
\(L(1)\) \(\approx\) \(1.054778044 + 0.4952408791i\)
\(L(1)\) \(\approx\) \(1.054778044 + 0.4952408791i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.587 + 0.809i)T \)
3 \( 1 + (0.707 + 0.707i)T \)
5 \( 1 + (0.951 - 0.309i)T \)
11 \( 1 + (0.891 - 0.453i)T \)
13 \( 1 + (-0.156 - 0.987i)T \)
17 \( 1 + (-0.453 - 0.891i)T \)
19 \( 1 + (-0.156 + 0.987i)T \)
23 \( 1 + (0.809 + 0.587i)T \)
29 \( 1 + (0.453 - 0.891i)T \)
31 \( 1 + (0.309 - 0.951i)T \)
37 \( 1 + (0.309 + 0.951i)T \)
43 \( 1 + (-0.587 + 0.809i)T \)
47 \( 1 + (-0.987 + 0.156i)T \)
53 \( 1 + (-0.453 + 0.891i)T \)
59 \( 1 + (0.809 + 0.587i)T \)
61 \( 1 + (-0.587 - 0.809i)T \)
67 \( 1 + (-0.891 - 0.453i)T \)
71 \( 1 + (-0.891 + 0.453i)T \)
73 \( 1 + iT \)
79 \( 1 + (0.707 + 0.707i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.987 - 0.156i)T \)
97 \( 1 + (-0.891 - 0.453i)T \)
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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

−25.495670978104503433373916481503, −24.83092711864600768365351647266, −23.65121318073058261188000214574, −22.32332397691854675038274064415, −21.54583357965075680046202740495, −20.77194933901534509377193671887, −19.62761477777302621006936233997, −19.260851471820474663301595385137, −18.051665058230833856172441409049, −17.579532130000170650809218439620, −16.61049878589727296712129532772, −14.87787095337849370242336873569, −14.06570634061755982439489049348, −13.15632898625496501607675482889, −12.39047458012398849417350094997, −11.28205105668101192583924354294, −10.15829109256126729428468342350, −9.08129121978870141565837823834, −8.71648042002826616357773000781, −7.06558732973469640685816823079, −6.58931668441157489979708552275, −4.56047307042047219579255404325, −3.2447311780505863602812643680, −2.15181689597618344143047389815, −1.42083186248739837771166301901, 1.35607854807781771571322571056, 2.818292004963779356802718165482, 4.42232184890487729319869160111, 5.42880430147072171128450050875, 6.39214965281218195340494698635, 7.798604685017146388401746051772, 8.66470279666213789508414577196, 9.57122493786647171298588036705, 10.0757797279722596301649320839, 11.30211827211295422492578772183, 13.17911069918838652220688409159, 13.88724491433215152265684960338, 14.73672322864438255729669214103, 15.5747076992180200092202397521, 16.63803414994001278544973191019, 17.18798501121144492263418223583, 18.28599702137425885899211861813, 19.32258046284217396575241275556, 20.20710386140081983374174362963, 21.037558822809710293173350128870, 22.178299826453973852352754649984, 22.87750786007005288012864020920, 24.49787866369283704815732734240, 24.98813496933596522775622372838, 25.49927165883226260737944309893

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