Properties

Label 1-287-287.130-r1-0-0
Degree $1$
Conductor $287$
Sign $0.00557 + 0.999i$
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 + (−0.258 + 0.965i)3-s + (−0.669 + 0.743i)4-s + (0.207 − 0.978i)5-s + (0.987 − 0.156i)6-s + (0.951 + 0.309i)8-s + (−0.866 − 0.5i)9-s + (−0.978 + 0.207i)10-s + (−0.838 − 0.544i)11-s + (−0.544 − 0.838i)12-s + (0.156 + 0.987i)13-s + (0.891 + 0.453i)15-s + (−0.104 − 0.994i)16-s + (0.544 − 0.838i)17-s + (−0.104 + 0.994i)18-s + (0.777 + 0.629i)19-s + ⋯
L(s)  = 1  + (−0.406 − 0.913i)2-s + (−0.258 + 0.965i)3-s + (−0.669 + 0.743i)4-s + (0.207 − 0.978i)5-s + (0.987 − 0.156i)6-s + (0.951 + 0.309i)8-s + (−0.866 − 0.5i)9-s + (−0.978 + 0.207i)10-s + (−0.838 − 0.544i)11-s + (−0.544 − 0.838i)12-s + (0.156 + 0.987i)13-s + (0.891 + 0.453i)15-s + (−0.104 − 0.994i)16-s + (0.544 − 0.838i)17-s + (−0.104 + 0.994i)18-s + (0.777 + 0.629i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00557 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.00557 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.00557 + 0.999i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (130, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ 0.00557 + 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3386570612 + 0.3405520141i\)
\(L(\frac12)\) \(\approx\) \(0.3386570612 + 0.3405520141i\)
\(L(1)\) \(\approx\) \(0.6431356381 - 0.1061010239i\)
\(L(1)\) \(\approx\) \(0.6431356381 - 0.1061010239i\)

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 \( 1 + (-0.258 + 0.965i)T \)
5 \( 1 + (0.207 - 0.978i)T \)
11 \( 1 + (-0.838 - 0.544i)T \)
13 \( 1 + (0.156 + 0.987i)T \)
17 \( 1 + (0.544 - 0.838i)T \)
19 \( 1 + (0.777 + 0.629i)T \)
23 \( 1 + (-0.913 + 0.406i)T \)
29 \( 1 + (0.453 - 0.891i)T \)
31 \( 1 + (0.978 - 0.207i)T \)
37 \( 1 + (-0.978 - 0.207i)T \)
43 \( 1 + (-0.587 + 0.809i)T \)
47 \( 1 + (-0.358 + 0.933i)T \)
53 \( 1 + (0.998 - 0.0523i)T \)
59 \( 1 + (-0.104 + 0.994i)T \)
61 \( 1 + (-0.994 + 0.104i)T \)
67 \( 1 + (0.0523 + 0.998i)T \)
71 \( 1 + (-0.891 + 0.453i)T \)
73 \( 1 + (-0.866 + 0.5i)T \)
79 \( 1 + (-0.965 + 0.258i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.629 + 0.777i)T \)
97 \( 1 + (0.891 + 0.453i)T \)
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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

−25.1803070698006059815602606747, −24.22491410414153858296075129136, −23.34813878062337008892096538722, −22.771486765657648622661141426605, −21.878144078556746651965079615544, −20.18708900016294726592585660872, −19.27525358020095558379392580211, −18.282359238185979107089864164794, −17.962137403813594664228245400563, −17.11576549471131996800452160903, −15.81328231033960335565482546232, −14.984214971092794277211216264731, −13.99184308170262528620386335886, −13.25350629540328853176474755687, −12.113257102139147788069545575271, −10.616492003429307188166829065881, −10.1408661462437852131891295713, −8.47744572727447151241121219390, −7.673471074202625340485835580716, −6.89661219147901744064401520884, −5.96267545134131449219616813425, −5.096561593351348453820337530394, −3.12287290982402346101951400468, −1.729247965650888084209404712459, −0.196370610109167019686885758429, 1.13261143840415797251159379962, 2.7081875155643154084793534976, 3.931975571785237648060945511269, 4.842240790211379879401631674591, 5.80089657535939270192078695645, 7.84263463532026247820259775439, 8.74709654014244813220295854782, 9.66360626043360778494334743182, 10.25039974870573953766028426303, 11.628614540273510675878145545229, 11.990642240996308189320870861633, 13.44997618204075357255471176573, 14.11175221708152430513204996039, 15.95554245168117019086047127166, 16.31254052438954067390055059228, 17.29355685323808946937784002783, 18.230355532289095277148954127725, 19.32846154622440203025647664075, 20.38063141529695448568765742726, 21.07459617443070014636713430548, 21.436375518999634781333390159285, 22.62449603476555580125619673230, 23.4682286054677264082581744133, 24.691225047028727329518295952435, 25.97268177514783788527206354924

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