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Properties

Label 1-287-287.186-r1-0-0
Degree $1$
Conductor $287$
Sign $0.926 + 0.377i$
Analytic cond. $30.8424$
Root an. cond. $30.8424$
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  + (0.406 + 0.913i)2-s + (−0.965 − 0.258i)3-s + (−0.669 + 0.743i)4-s + (−0.207 + 0.978i)5-s + (−0.156 − 0.987i)6-s + (−0.951 − 0.309i)8-s + (0.866 + 0.5i)9-s + (−0.978 + 0.207i)10-s + (0.544 − 0.838i)11-s + (0.838 − 0.544i)12-s + (−0.987 + 0.156i)13-s + (0.453 − 0.891i)15-s + (−0.104 − 0.994i)16-s + (−0.838 − 0.544i)17-s + (−0.104 + 0.994i)18-s + (0.629 − 0.777i)19-s + ⋯
L(s)  = 1  + (0.406 + 0.913i)2-s + (−0.965 − 0.258i)3-s + (−0.669 + 0.743i)4-s + (−0.207 + 0.978i)5-s + (−0.156 − 0.987i)6-s + (−0.951 − 0.309i)8-s + (0.866 + 0.5i)9-s + (−0.978 + 0.207i)10-s + (0.544 − 0.838i)11-s + (0.838 − 0.544i)12-s + (−0.987 + 0.156i)13-s + (0.453 − 0.891i)15-s + (−0.104 − 0.994i)16-s + (−0.838 − 0.544i)17-s + (−0.104 + 0.994i)18-s + (0.629 − 0.777i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.926 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.926 + 0.377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.926 + 0.377i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (186, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.926 + 0.377i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9568954911 + 0.1873948702i\)
\(L(\frac12)\) \(\approx\) \(0.9568954911 + 0.1873948702i\)
\(L(1)\) \(\approx\) \(0.7122881601 + 0.3637133656i\)
\(L(1)\) \(\approx\) \(0.7122881601 + 0.3637133656i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.406 + 0.913i)T \)
3 \( 1 + (-0.965 - 0.258i)T \)
5 \( 1 + (-0.207 + 0.978i)T \)
11 \( 1 + (0.544 - 0.838i)T \)
13 \( 1 + (-0.987 + 0.156i)T \)
17 \( 1 + (-0.838 - 0.544i)T \)
19 \( 1 + (0.629 - 0.777i)T \)
23 \( 1 + (-0.913 + 0.406i)T \)
29 \( 1 + (0.891 + 0.453i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (0.587 - 0.809i)T \)
47 \( 1 + (0.933 + 0.358i)T \)
53 \( 1 + (0.0523 + 0.998i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + (0.994 - 0.104i)T \)
67 \( 1 + (0.998 - 0.0523i)T \)
71 \( 1 + (-0.453 - 0.891i)T \)
73 \( 1 + (0.866 - 0.5i)T \)
79 \( 1 + (-0.258 - 0.965i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.777 - 0.629i)T \)
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

−24.74650122692327069736126645489, −24.210993495787663672820691624363, −23.18330956117846952737538724238, −22.4839225105422766272602389419, −21.719655906116392024526180584455, −20.73218478566767089621133865421, −20.0131015760903423938557797420, −19.111608008259088252301979643464, −17.70693322571626979056715730874, −17.322545165387065410155656926969, −16.03191178386618970274302416091, −15.13651468026250819945317592557, −13.93177957544280764440455271474, −12.57839466882857431921637418892, −12.27610783117843433268066824203, −11.44554395466056782216351134274, −10.118664087535542556712015164061, −9.64419102561741416272946982467, −8.30481492339916335006556387194, −6.66187621242158093627856944756, −5.47214157517132026978141394755, −4.610632709314204373803154812587, −3.94278709977941206347602604556, −2.0466394238752990320266605478, −0.857645128924078776219318792063, 0.40719992245864640276796722701, 2.64862382936652168992927799497, 4.03238127046753142275896982958, 5.116119523084218245246439975776, 6.20936409622375578261974971771, 6.92038849868985061544090829994, 7.68005840925089630044945151488, 9.13714151729240793196384844676, 10.40021775713322990616626712819, 11.60116147461474905391909583563, 12.139731189532824734400787352999, 13.61337495428177316530712120800, 14.12999125619266495545628064400, 15.4759136638975327661045836607, 16.01061812803302728453636902004, 17.21549295402003977495987936589, 17.76632656793383997042472793528, 18.69741184303194968891124724695, 19.6361603939717941107206317405, 21.46997278824596850543772816441, 22.24375559728637586163527087534, 22.47031269340938943825554851640, 23.69461976938990267669827415443, 24.264666919029557803067308762971, 25.09443562441066221163716896759

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