Properties

Label 1-287-287.142-r1-0-0
Degree $1$
Conductor $287$
Sign $-0.890 + 0.455i$
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  + (−0.994 − 0.104i)2-s + (−0.258 − 0.965i)3-s + (0.978 + 0.207i)4-s + (−0.743 − 0.669i)5-s + (0.156 + 0.987i)6-s + (−0.951 − 0.309i)8-s + (−0.866 + 0.5i)9-s + (0.669 + 0.743i)10-s + (0.998 + 0.0523i)11-s + (−0.0523 − 0.998i)12-s + (0.987 − 0.156i)13-s + (−0.453 + 0.891i)15-s + (0.913 + 0.406i)16-s + (0.0523 − 0.998i)17-s + (0.913 − 0.406i)18-s + (−0.358 − 0.933i)19-s + ⋯
L(s)  = 1  + (−0.994 − 0.104i)2-s + (−0.258 − 0.965i)3-s + (0.978 + 0.207i)4-s + (−0.743 − 0.669i)5-s + (0.156 + 0.987i)6-s + (−0.951 − 0.309i)8-s + (−0.866 + 0.5i)9-s + (0.669 + 0.743i)10-s + (0.998 + 0.0523i)11-s + (−0.0523 − 0.998i)12-s + (0.987 − 0.156i)13-s + (−0.453 + 0.891i)15-s + (0.913 + 0.406i)16-s + (0.0523 − 0.998i)17-s + (0.913 − 0.406i)18-s + (−0.358 − 0.933i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.890 + 0.455i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.890 + 0.455i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (142, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ -0.890 + 0.455i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.1435262063 - 0.5952105969i\)
\(L(\frac12)\) \(\approx\) \(-0.1435262063 - 0.5952105969i\)
\(L(1)\) \(\approx\) \(0.4512171842 - 0.3497675389i\)
\(L(1)\) \(\approx\) \(0.4512171842 - 0.3497675389i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.994 + 0.104i)T \)
3 \( 1 + (0.258 + 0.965i)T \)
5 \( 1 + (0.743 + 0.669i)T \)
11 \( 1 + (-0.998 - 0.0523i)T \)
13 \( 1 + (-0.987 + 0.156i)T \)
17 \( 1 + (-0.0523 + 0.998i)T \)
19 \( 1 + (0.358 + 0.933i)T \)
23 \( 1 + (-0.104 + 0.994i)T \)
29 \( 1 + (0.891 + 0.453i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (-0.669 + 0.743i)T \)
43 \( 1 + (-0.587 + 0.809i)T \)
47 \( 1 + (-0.777 + 0.629i)T \)
53 \( 1 + (0.838 - 0.544i)T \)
59 \( 1 + (-0.913 + 0.406i)T \)
61 \( 1 + (0.406 - 0.913i)T \)
67 \( 1 + (-0.544 - 0.838i)T \)
71 \( 1 + (-0.453 - 0.891i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (0.965 + 0.258i)T \)
83 \( 1 - T \)
89 \( 1 + (0.933 - 0.358i)T \)
97 \( 1 + (0.453 - 0.891i)T \)
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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

−25.9019522354143117091105672487, −25.45137503283083142627420746064, −23.918734080487215185621138561394, −23.19854341486787111689565947332, −22.11942257038363314578596674436, −21.2274359388943865318687715527, −20.21444359043720619800561772420, −19.43456037061881621246890286555, −18.56378819527180534529954776286, −17.494383722052072989295847475354, −16.6252839966431594921734062846, −15.88150491971373104114639212000, −14.9870994641556585198814282484, −14.36254812721282295773599096770, −12.33034339894284773649868200785, −11.26154059862020534789025362711, −10.880504107047379100439312787033, −9.78055936553912834629604283385, −8.826968375175941513037666248614, −7.91936491954033666569201095968, −6.5963690327465308329957814545, −5.82344980035222854261804559405, −3.98543515167084789125859133815, −3.29144448014153378507411717810, −1.444583294349259409632505817528, 0.32440449041285129784604867251, 1.11209197136378464581918392610, 2.45553128379036049956741635882, 3.9705217275984219279526551847, 5.6721578423041151650095577679, 6.798832152010208408883635316208, 7.57499656714065030461426668939, 8.62608076042990912852925091501, 9.22457244491111719495903974524, 10.979178152678639115040413814148, 11.52953197131948380742450296248, 12.40772582475764762252287479423, 13.32884968151744129705309687695, 14.76340953324604691864772859168, 15.95587881823042881756622886201, 16.72363348731547563737769565651, 17.497918739882601843308255109780, 18.54249188106719400581974037583, 19.13972736152215345001497975837, 20.1486063378588500684199014292, 20.57551837789948783010697544564, 22.18015547662434959395824988771, 23.24375564112120645786631575456, 24.139422982920255292165648423206, 24.82157159085591052792696865835

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