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Properties

Label 1-287-287.33-r1-0-0
Degree $1$
Conductor $287$
Sign $-0.999 + 0.00376i$
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  + (−0.669 + 0.743i)2-s + (−0.866 − 0.5i)3-s + (−0.104 − 0.994i)4-s + (0.913 + 0.406i)5-s + (0.951 − 0.309i)6-s + (0.809 + 0.587i)8-s + (0.5 + 0.866i)9-s + (−0.913 + 0.406i)10-s + (−0.406 − 0.913i)11-s + (−0.406 + 0.913i)12-s + (−0.951 + 0.309i)13-s + (−0.587 − 0.809i)15-s + (−0.978 + 0.207i)16-s + (−0.406 − 0.913i)17-s + (−0.978 − 0.207i)18-s + (0.207 + 0.978i)19-s + ⋯
L(s)  = 1  + (−0.669 + 0.743i)2-s + (−0.866 − 0.5i)3-s + (−0.104 − 0.994i)4-s + (0.913 + 0.406i)5-s + (0.951 − 0.309i)6-s + (0.809 + 0.587i)8-s + (0.5 + 0.866i)9-s + (−0.913 + 0.406i)10-s + (−0.406 − 0.913i)11-s + (−0.406 + 0.913i)12-s + (−0.951 + 0.309i)13-s + (−0.587 − 0.809i)15-s + (−0.978 + 0.207i)16-s + (−0.406 − 0.913i)17-s + (−0.978 − 0.207i)18-s + (0.207 + 0.978i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00376i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.00376i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.999 + 0.00376i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ -0.999 + 0.00376i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.0003611845145 + 0.1916422530i\)
\(L(\frac12)\) \(\approx\) \(0.0003611845145 + 0.1916422530i\)
\(L(1)\) \(\approx\) \(0.5424595768 + 0.1189933458i\)
\(L(1)\) \(\approx\) \(0.5424595768 + 0.1189933458i\)

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 \( 1 + (-0.866 - 0.5i)T \)
5 \( 1 + (0.913 + 0.406i)T \)
11 \( 1 + (-0.406 - 0.913i)T \)
13 \( 1 + (-0.951 + 0.309i)T \)
17 \( 1 + (-0.406 - 0.913i)T \)
19 \( 1 + (0.207 + 0.978i)T \)
23 \( 1 + (0.669 - 0.743i)T \)
29 \( 1 + (0.587 + 0.809i)T \)
31 \( 1 + (-0.913 + 0.406i)T \)
37 \( 1 + (0.913 + 0.406i)T \)
43 \( 1 + (-0.309 - 0.951i)T \)
47 \( 1 + (-0.743 - 0.669i)T \)
53 \( 1 + (-0.994 + 0.104i)T \)
59 \( 1 + (0.978 + 0.207i)T \)
61 \( 1 + (-0.978 + 0.207i)T \)
67 \( 1 + (0.994 - 0.104i)T \)
71 \( 1 + (-0.587 + 0.809i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.866 + 0.5i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.207 - 0.978i)T \)
97 \( 1 + (0.587 + 0.809i)T \)
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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

−25.05878097576873555691955689347, −23.912092677573669653160623748042, −22.71433547069882116102697958859, −21.851637958614665244324972415493, −21.34060207889070969889583705534, −20.39845959166819006136577487217, −19.52823324938287514631280223242, −18.1110678747708004810720875280, −17.51278859787703465893816172939, −17.0646881914101196018440436500, −15.93579953553675140555601629426, −14.85974694402888458483381604872, −13.11334166653435092959690691061, −12.69646074437261715747758818290, −11.54245290946474212180653459526, −10.59998071059030545677899406094, −9.73997261015386098898244963687, −9.2351985221759057065369712006, −7.70763049649175029382061332283, −6.52538282520832924656568155274, −5.1518930145393942275542452818, −4.36222078240647261747412254562, −2.71389149982457409073019769853, −1.49499708090205260202437033780, −0.0868368114568335344154936206, 1.27823167907568958663248985586, 2.54770891187472235056429700508, 4.92733087967392791037491605401, 5.62487064927516884605693240610, 6.63022069648004436784275209933, 7.2948483113925794234993448967, 8.578185701804994388036202073393, 9.78278312873380490816255225793, 10.56858262029157203074051679292, 11.45897531756124105039702154086, 12.86770806167613342496919459950, 13.8842264123917152176426150107, 14.614556382396534194898918040284, 16.082701524004687816846518679659, 16.7034412065504622525351898468, 17.49998933785213863824108150903, 18.508150262249785354933221083473, 18.71206226659822865533489647875, 20.08303002972406361904013198689, 21.51013255845493579489416403869, 22.3303937376244427298337926518, 23.197365585394040391197202834862, 24.200085576192774512354042031898, 24.81621011516339856831853247938, 25.563347949272997206110402808758

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