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Properties

Label 1-287-287.242-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: $\chi_{287} (242, \cdot )$
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

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

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