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Properties

Label 1-287-287.152-r0-0-0
Degree $1$
Conductor $287$
Sign $0.581 + 0.813i$
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  + (0.207 + 0.978i)2-s + (−0.965 + 0.258i)3-s + (−0.913 + 0.406i)4-s + (0.994 + 0.104i)5-s + (−0.453 − 0.891i)6-s + (−0.587 − 0.809i)8-s + (0.866 − 0.5i)9-s + (0.104 + 0.994i)10-s + (0.629 − 0.777i)11-s + (0.777 − 0.629i)12-s + (0.891 − 0.453i)13-s + (−0.987 + 0.156i)15-s + (0.669 − 0.743i)16-s + (−0.777 − 0.629i)17-s + (0.669 + 0.743i)18-s + (−0.0523 − 0.998i)19-s + ⋯
L(s)  = 1  + (0.207 + 0.978i)2-s + (−0.965 + 0.258i)3-s + (−0.913 + 0.406i)4-s + (0.994 + 0.104i)5-s + (−0.453 − 0.891i)6-s + (−0.587 − 0.809i)8-s + (0.866 − 0.5i)9-s + (0.104 + 0.994i)10-s + (0.629 − 0.777i)11-s + (0.777 − 0.629i)12-s + (0.891 − 0.453i)13-s + (−0.987 + 0.156i)15-s + (0.669 − 0.743i)16-s + (−0.777 − 0.629i)17-s + (0.669 + 0.743i)18-s + (−0.0523 − 0.998i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.581 + 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.581 + 0.813i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (152, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.581 + 0.813i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.009066770 + 0.5188971767i\)
\(L(\frac12)\) \(\approx\) \(1.009066770 + 0.5188971767i\)
\(L(1)\) \(\approx\) \(0.9066879643 + 0.4327462116i\)
\(L(1)\) \(\approx\) \(0.9066879643 + 0.4327462116i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.207 + 0.978i)T \)
3 \( 1 + (-0.965 + 0.258i)T \)
5 \( 1 + (0.994 + 0.104i)T \)
11 \( 1 + (0.629 - 0.777i)T \)
13 \( 1 + (0.891 - 0.453i)T \)
17 \( 1 + (-0.777 - 0.629i)T \)
19 \( 1 + (-0.0523 - 0.998i)T \)
23 \( 1 + (0.978 - 0.207i)T \)
29 \( 1 + (0.156 + 0.987i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (-0.951 - 0.309i)T \)
47 \( 1 + (-0.544 + 0.838i)T \)
53 \( 1 + (0.933 + 0.358i)T \)
59 \( 1 + (-0.669 - 0.743i)T \)
61 \( 1 + (0.743 + 0.669i)T \)
67 \( 1 + (-0.358 + 0.933i)T \)
71 \( 1 + (0.987 + 0.156i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (0.258 - 0.965i)T \)
83 \( 1 - T \)
89 \( 1 + (0.998 - 0.0523i)T \)
97 \( 1 + (0.987 - 0.156i)T \)
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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

−25.23016277811430415810629787771, −24.37227602624636743190600201462, −23.10875964806477414770006590419, −22.79581599023540777640416919363, −21.54940300323540987979401205760, −21.2416090056003711399488954756, −20.02101722296824330804839548483, −18.943352951683911974507360891861, −18.06358893574353098829682249151, −17.44736699173548638834728871572, −16.58381514023492307624078421153, −15.06236333860827718442632663023, −13.8992225611571960575905104942, −13.08569955759999206597720120385, −12.334610417024686735705454176924, −11.33445328362322063430999218845, −10.47598508681356226093377466365, −9.6467961210058126787616218004, −8.59365233547729587392867574424, −6.75765269348791632169932788436, −5.91848976423150247395960813184, −4.893320889077348687901433922108, −3.84932411267763783205096664233, −2.02880921295964704062675366601, −1.33776751988161497971924327527, 0.98656390314269339893235794925, 3.1879636064442960945773139641, 4.594149822234073736815295187199, 5.47037631565971114962748224010, 6.35903687586924542929333718749, 6.95878471432597494536864360069, 8.688471922336630640818419962420, 9.3986035084229133276993347952, 10.62974867555587119169095555880, 11.5785545446716007270693289600, 13.06283104215217701386006344705, 13.488428680742197595363232474, 14.7341890140325207990785845075, 15.6995205034052179143427987130, 16.57570810656405653635851147840, 17.29477920943466504030584747322, 18.03382347518712139001982462859, 18.77826564749847414228854156713, 20.57080816182933946377802878962, 21.642574591352757842649936605930, 22.15170309074405627282081950429, 22.907991902447277772320674606192, 23.966980383223581218434038033438, 24.65597418435389230951704169089, 25.55919891632826692839907192671

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