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Properties

Label 1-287-287.25-r0-0-0
Degree $1$
Conductor $287$
Sign $0.999 - 0.0377i$
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  + (0.913 − 0.406i)2-s + (0.5 + 0.866i)3-s + (0.669 − 0.743i)4-s + (−0.978 − 0.207i)5-s + (0.809 + 0.587i)6-s + (0.309 − 0.951i)8-s + (−0.5 + 0.866i)9-s + (−0.978 + 0.207i)10-s + (0.978 − 0.207i)11-s + (0.978 + 0.207i)12-s + (0.809 + 0.587i)13-s + (−0.309 − 0.951i)15-s + (−0.104 − 0.994i)16-s + (0.978 − 0.207i)17-s + (−0.104 + 0.994i)18-s + (0.104 + 0.994i)19-s + ⋯
L(s)  = 1  + (0.913 − 0.406i)2-s + (0.5 + 0.866i)3-s + (0.669 − 0.743i)4-s + (−0.978 − 0.207i)5-s + (0.809 + 0.587i)6-s + (0.309 − 0.951i)8-s + (−0.5 + 0.866i)9-s + (−0.978 + 0.207i)10-s + (0.978 − 0.207i)11-s + (0.978 + 0.207i)12-s + (0.809 + 0.587i)13-s + (−0.309 − 0.951i)15-s + (−0.104 − 0.994i)16-s + (0.978 − 0.207i)17-s + (−0.104 + 0.994i)18-s + (0.104 + 0.994i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0377i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0377i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.999 - 0.0377i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.999 - 0.0377i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.292921914 - 0.04328650525i\)
\(L(\frac12)\) \(\approx\) \(2.292921914 - 0.04328650525i\)
\(L(1)\) \(\approx\) \(1.861459429 - 0.05760145618i\)
\(L(1)\) \(\approx\) \(1.861459429 - 0.05760145618i\)

Euler product

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

−25.44808791187608096266703505307, −24.484012124838595949574024853826, −23.6874804640819168979163365439, −23.07693493170721133023954053445, −22.24717100526477870114022338119, −20.94591981154286188631285669602, −20.04544601382309720056869442776, −19.42912755552465369609906756036, −18.287805987983303926788010202878, −17.20796565361752985388621661969, −16.13783983799178327066730876653, −15.02351818589728977594196983518, −14.596355107867432865984312329, −13.43610455627935405930913949775, −12.64245383134776092891009363525, −11.784777326698953609432433042149, −10.99774727286716293902049691929, −9.00206540450883186944446470838, −8.068044142340180287170636935891, −7.18899894475338738270681203241, −6.48670026200399751998497736894, −5.142360840999934125723962888595, −3.627014827537887499803425522581, −3.1710490820553209844576761521, −1.45847765122889884103487873274, 1.52442752857393619756958091422, 3.2738048765501432888690714622, 3.78601017494734831531034115985, 4.7182585405853465831194063672, 5.89873911313310032762674786653, 7.27604249754219513733165625182, 8.53459244772433740583748649027, 9.544449104169751574552535447114, 10.72896856965121698963970909646, 11.5173846130173880284292552083, 12.3436280886081249116144177782, 13.65684777231070145385280947285, 14.50009519467976979378461471708, 15.14751814443212391018812156536, 16.252185316725322155511982023383, 16.64760060571934972476826179971, 18.90052998739620943337808096327, 19.28102219328925702732264575733, 20.47936751651997544623762932413, 20.83067692997192652680016344626, 21.894974950603456563202698255669, 22.82077942540976780419200984139, 23.385856201033079381191084780684, 24.62292721900098197464987874367, 25.30060835440692055659351187207

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