Properties

Label 1-287-287.39-r0-0-0
Degree $1$
Conductor $287$
Sign $0.859 - 0.511i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.104 + 0.994i)2-s + (0.866 − 0.5i)3-s + (−0.978 + 0.207i)4-s + (−0.669 − 0.743i)5-s + (0.587 + 0.809i)6-s + (−0.309 − 0.951i)8-s + (0.5 − 0.866i)9-s + (0.669 − 0.743i)10-s + (0.743 + 0.669i)11-s + (−0.743 + 0.669i)12-s + (−0.587 − 0.809i)13-s + (−0.951 − 0.309i)15-s + (0.913 − 0.406i)16-s + (−0.743 − 0.669i)17-s + (0.913 + 0.406i)18-s + (−0.406 − 0.913i)19-s + ⋯
L(s)  = 1  + (0.104 + 0.994i)2-s + (0.866 − 0.5i)3-s + (−0.978 + 0.207i)4-s + (−0.669 − 0.743i)5-s + (0.587 + 0.809i)6-s + (−0.309 − 0.951i)8-s + (0.5 − 0.866i)9-s + (0.669 − 0.743i)10-s + (0.743 + 0.669i)11-s + (−0.743 + 0.669i)12-s + (−0.587 − 0.809i)13-s + (−0.951 − 0.309i)15-s + (0.913 − 0.406i)16-s + (−0.743 − 0.669i)17-s + (0.913 + 0.406i)18-s + (−0.406 − 0.913i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.859 - 0.511i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.859 - 0.511i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.859 - 0.511i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.227392889 - 0.3376801147i\)
\(L(\frac12)\) \(\approx\) \(1.227392889 - 0.3376801147i\)
\(L(1)\) \(\approx\) \(1.149948347 + 0.04189344648i\)
\(L(1)\) \(\approx\) \(1.149948347 + 0.04189344648i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.104 + 0.994i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-0.669 - 0.743i)T \)
11 \( 1 + (0.743 + 0.669i)T \)
13 \( 1 + (-0.587 - 0.809i)T \)
17 \( 1 + (-0.743 - 0.669i)T \)
19 \( 1 + (-0.406 - 0.913i)T \)
23 \( 1 + (-0.104 - 0.994i)T \)
29 \( 1 + (0.951 + 0.309i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (0.669 + 0.743i)T \)
43 \( 1 + (0.809 - 0.587i)T \)
47 \( 1 + (-0.994 + 0.104i)T \)
53 \( 1 + (-0.207 - 0.978i)T \)
59 \( 1 + (0.913 + 0.406i)T \)
61 \( 1 + (-0.913 + 0.406i)T \)
67 \( 1 + (0.207 + 0.978i)T \)
71 \( 1 + (-0.951 + 0.309i)T \)
73 \( 1 + (0.5 + 0.866i)T \)
79 \( 1 + (-0.866 - 0.5i)T \)
83 \( 1 + T \)
89 \( 1 + (0.406 + 0.913i)T \)
97 \( 1 + (-0.951 - 0.309i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.97761880493914512254449257307, −24.691166768795498953088880425436, −23.61096443816337281355345953541, −22.61674639598426881108403553454, −21.668006853670106364289906248333, −21.318217225181451707973598939661, −19.87158788706084484682621526120, −19.45311496765839703868359820168, −18.883865069899455742968981112369, −17.60498170905928866936039702699, −16.32592839402624883565145840833, −15.14590630267463872360029120538, −14.34762051204623712335838779316, −13.76503839646941198785854834398, −12.41816681721858977518720643001, −11.42866243835701536406623601205, −10.61917645715175754303591395305, −9.6496394365317647425187487128, −8.70219745473180131623886093012, −7.79676867630884505408199257613, −6.30012319622722620023504436818, −4.53040877643461139411634887220, −3.838153625908184573401992776343, −2.924920399998728322333961820673, −1.7661795987867344788335048260, 0.78589297425580372315701986829, 2.69950776338671606718170655654, 4.15532123367430990566198318000, 4.83548681540446949386355647154, 6.508307715861234420247663681350, 7.28996131846159195043857161758, 8.2461581260673949316097836723, 8.93672472785804050368239435586, 9.871828653749903848206609530976, 11.85023223461098504581752002526, 12.72183923571887919291279960389, 13.40349277348941687934023959908, 14.59227754366647887281624383733, 15.2214946627246889023223616494, 16.05386516595110120540975101194, 17.265828671934438039748126094030, 17.93836457755810055364952677781, 19.173377315458366516473670714101, 19.90494439207591430070632800149, 20.71289041875881385665354141750, 22.12667251644294694391296615597, 22.98311799010601175467693050658, 24.01042970587047301100113871694, 24.60137658799072592684861543736, 25.20569756936760788851254668676

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