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Properties

Label 1-287-287.51-r0-0-0
Degree $1$
Conductor $287$
Sign $0.909 - 0.415i$
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  + (0.669 − 0.743i)2-s + (−0.5 + 0.866i)3-s + (−0.104 − 0.994i)4-s + (0.913 + 0.406i)5-s + (0.309 + 0.951i)6-s + (−0.809 − 0.587i)8-s + (−0.5 − 0.866i)9-s + (0.913 − 0.406i)10-s + (0.913 − 0.406i)11-s + (0.913 + 0.406i)12-s + (0.309 + 0.951i)13-s + (−0.809 + 0.587i)15-s + (−0.978 + 0.207i)16-s + (0.913 − 0.406i)17-s + (−0.978 − 0.207i)18-s + (−0.978 + 0.207i)19-s + ⋯
L(s)  = 1  + (0.669 − 0.743i)2-s + (−0.5 + 0.866i)3-s + (−0.104 − 0.994i)4-s + (0.913 + 0.406i)5-s + (0.309 + 0.951i)6-s + (−0.809 − 0.587i)8-s + (−0.5 − 0.866i)9-s + (0.913 − 0.406i)10-s + (0.913 − 0.406i)11-s + (0.913 + 0.406i)12-s + (0.309 + 0.951i)13-s + (−0.809 + 0.587i)15-s + (−0.978 + 0.207i)16-s + (0.913 − 0.406i)17-s + (−0.978 − 0.207i)18-s + (−0.978 + 0.207i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.909 - 0.415i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (51, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.909 - 0.415i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.712559069 - 0.3726045988i\)
\(L(\frac12)\) \(\approx\) \(1.712559069 - 0.3726045988i\)
\(L(1)\) \(\approx\) \(1.439969595 - 0.2551493961i\)
\(L(1)\) \(\approx\) \(1.439969595 - 0.2551493961i\)

Euler product

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

−25.125472495453360739923145314737, −24.93077937401282204169880710842, −23.7487196352894202949477684988, −23.05989559448459109763488215404, −22.20944326057100865145029567125, −21.34620675904911229969844042786, −20.331268013351724203082406587748, −19.070839163914087845257548049222, −17.8690272207008201145973685758, −17.200456616864293258776963937619, −16.77600238318501965965572133033, −15.36110138583226985739189070439, −14.3404673535053004801691791846, −13.44716847455600337435230237714, −12.74275973103994846391662044917, −12.03395716370197810318448705968, −10.72954872747728043948765212235, −9.251659508707524668076336273291, −8.17222141114684597848578448471, −7.16430504150530423676909916992, −6.078882656603796439840041410322, −5.60761252908874255501698630412, −4.3574564800437186064735942523, −2.75614359308817484648928105055, −1.35569679180290139486927247141, 1.33439679232130633860653481425, 2.79557042572346641750788364837, 3.88510312354439386462259281991, 4.872206661950677476851851393416, 6.03032134985109957379240070112, 6.56513652658964142346121303923, 8.96651842453895409916218993056, 9.61141358394118208111160069648, 10.58628198256845064602386512807, 11.31107722349999195087042849474, 12.22786587419323505658862796728, 13.466901512711275355802052084365, 14.45683893066041485612770870700, 14.89098692207675418140458577548, 16.400643286233401306195382322234, 17.071504656034306811503145021213, 18.37558165264139782832431460936, 19.11309469797141457511065238009, 20.44995228686598025746265491057, 21.233394965728065205990689529907, 21.70531682794696870307360320229, 22.58878279983271439309016457026, 23.24900951169872016783375700579, 24.39268719002590509338612462425, 25.4682108673407449910295144276

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