Properties

Label 1-287-287.145-r0-0-0
Degree $1$
Conductor $287$
Sign $0.317 - 0.948i$
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  + (−0.994 − 0.104i)2-s + (−0.258 − 0.965i)3-s + (0.978 + 0.207i)4-s + (0.743 + 0.669i)5-s + (0.156 + 0.987i)6-s + (−0.951 − 0.309i)8-s + (−0.866 + 0.5i)9-s + (−0.669 − 0.743i)10-s + (−0.998 − 0.0523i)11-s + (−0.0523 − 0.998i)12-s + (0.987 − 0.156i)13-s + (0.453 − 0.891i)15-s + (0.913 + 0.406i)16-s + (0.0523 − 0.998i)17-s + (0.913 − 0.406i)18-s + (−0.358 − 0.933i)19-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)2-s + (−0.258 − 0.965i)3-s + (0.978 + 0.207i)4-s + (0.743 + 0.669i)5-s + (0.156 + 0.987i)6-s + (−0.951 − 0.309i)8-s + (−0.866 + 0.5i)9-s + (−0.669 − 0.743i)10-s + (−0.998 − 0.0523i)11-s + (−0.0523 − 0.998i)12-s + (0.987 − 0.156i)13-s + (0.453 − 0.891i)15-s + (0.913 + 0.406i)16-s + (0.0523 − 0.998i)17-s + (0.913 − 0.406i)18-s + (−0.358 − 0.933i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.317 - 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.317 - 0.948i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.317 - 0.948i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6252719436 - 0.4498862665i\)
\(L(\frac12)\) \(\approx\) \(0.6252719436 - 0.4498862665i\)
\(L(1)\) \(\approx\) \(0.6796381558 - 0.2336176849i\)
\(L(1)\) \(\approx\) \(0.6796381558 - 0.2336176849i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.994 + 0.104i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (-0.743 - 0.669i)T \)
11 \( 1 + (0.998 + 0.0523i)T \)
13 \( 1 + (-0.987 + 0.156i)T \)
17 \( 1 + (-0.0523 + 0.998i)T \)
19 \( 1 + (0.358 + 0.933i)T \)
23 \( 1 + (-0.104 + 0.994i)T \)
29 \( 1 + (-0.891 - 0.453i)T \)
31 \( 1 + (-0.669 - 0.743i)T \)
37 \( 1 + (-0.669 + 0.743i)T \)
43 \( 1 + (-0.587 + 0.809i)T \)
47 \( 1 + (-0.777 + 0.629i)T \)
53 \( 1 + (-0.838 + 0.544i)T \)
59 \( 1 + (0.913 - 0.406i)T \)
61 \( 1 + (-0.406 + 0.913i)T \)
67 \( 1 + (0.544 + 0.838i)T \)
71 \( 1 + (0.453 + 0.891i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (-0.965 - 0.258i)T \)
83 \( 1 + T \)
89 \( 1 + (0.933 - 0.358i)T \)
97 \( 1 + (0.453 - 0.891i)T \)
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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

−25.80481737835606766911865568083, −25.24636932648314074126553974630, −23.92986920368209521373176791334, −23.20815581944067636145496182577, −21.63798962624555198561067293089, −21.021536162642748364024430626900, −20.52715090692790798276742238532, −19.32456085737295623191687730257, −18.17375349703601606971395956183, −17.36895487499721018980624946926, −16.63678201491017117542681014201, −15.83719837842234220971856175285, −15.075221138866730688899164916632, −13.7380645522103540920808937807, −12.483623571384672908367347500645, −11.29147955132411286281193855662, −10.34825562914004492739355070155, −9.77572029456225200743844474405, −8.70006154757034187563048067325, −8.007321760985906607808311823772, −6.14500542573506664978038963085, −5.68884394396444875538583577511, −4.23908984333590346213132248882, −2.72126383201660973191961619063, −1.273603092591800750535634018323, 0.823840150797044886516225513103, 2.283369172506714005148591750426, 2.91853962217756006198100011276, 5.35360496158089461471954868393, 6.43348857831277565883950545749, 7.08032424422244468279114107896, 8.182065323332018274641222585223, 9.11886978142960697995672680340, 10.54410035766085478817278520753, 10.93270044893333488492115358322, 12.161429213521019466270720069181, 13.21667087155917046947388297445, 14.0704355694329692439439207655, 15.436972258549534556098493138580, 16.423665133396511932081040422872, 17.521089940495866650464388119593, 18.22431952301678893579064421088, 18.55797561741155380282120699414, 19.64826606772909462173472335380, 20.68673806051084289083265076010, 21.5483712599059840810116244205, 22.7991023196570626452415746319, 23.67368660997232568728534020322, 24.75014063274276583707036665185, 25.45800928916747732857167475453

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