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Properties

Label 1-287-287.174-r1-0-0
Degree $1$
Conductor $287$
Sign $-0.999 - 0.0317i$
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  + (0.309 + 0.951i)2-s − 3-s + (−0.809 + 0.587i)4-s + (0.809 − 0.587i)5-s + (−0.309 − 0.951i)6-s + (−0.809 − 0.587i)8-s + 9-s + (0.809 + 0.587i)10-s + (−0.809 − 0.587i)11-s + (0.809 − 0.587i)12-s + (−0.309 − 0.951i)13-s + (−0.809 + 0.587i)15-s + (0.309 − 0.951i)16-s + (0.809 + 0.587i)17-s + (0.309 + 0.951i)18-s + (−0.309 + 0.951i)19-s + ⋯
L(s)  = 1  + (0.309 + 0.951i)2-s − 3-s + (−0.809 + 0.587i)4-s + (0.809 − 0.587i)5-s + (−0.309 − 0.951i)6-s + (−0.809 − 0.587i)8-s + 9-s + (0.809 + 0.587i)10-s + (−0.809 − 0.587i)11-s + (0.809 − 0.587i)12-s + (−0.309 − 0.951i)13-s + (−0.809 + 0.587i)15-s + (0.309 − 0.951i)16-s + (0.809 + 0.587i)17-s + (0.309 + 0.951i)18-s + (−0.309 + 0.951i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0317i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.999 - 0.0317i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (174, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ -0.999 - 0.0317i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01039028552 + 0.6542971213i\)
\(L(\frac12)\) \(\approx\) \(0.01039028552 + 0.6542971213i\)
\(L(1)\) \(\approx\) \(0.7031533470 + 0.3752481439i\)
\(L(1)\) \(\approx\) \(0.7031533470 + 0.3752481439i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.309 + 0.951i)T \)
3 \( 1 - T \)
5 \( 1 + (0.809 - 0.587i)T \)
11 \( 1 + (-0.809 - 0.587i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
17 \( 1 + (0.809 + 0.587i)T \)
19 \( 1 + (-0.309 + 0.951i)T \)
23 \( 1 + (0.309 + 0.951i)T \)
29 \( 1 + (-0.809 + 0.587i)T \)
31 \( 1 + (0.809 + 0.587i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
43 \( 1 + (0.309 + 0.951i)T \)
47 \( 1 + (-0.309 - 0.951i)T \)
53 \( 1 + (-0.809 + 0.587i)T \)
59 \( 1 + (-0.309 - 0.951i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 + (-0.809 + 0.587i)T \)
71 \( 1 + (-0.809 - 0.587i)T \)
73 \( 1 - T \)
79 \( 1 + T \)
83 \( 1 - T \)
89 \( 1 + (-0.309 + 0.951i)T \)
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

−24.53167966583141321875949185909, −23.65705841994076925835899633416, −22.73994542166473740237384056558, −22.23411398854929234640581959928, −21.16965881549702894911759634379, −20.8109841812901023495036827190, −19.11290947499132897074087336814, −18.569892561500664882436003432563, −17.690544555022242134707271557, −16.93743097902554298073880470158, −15.51974098782785710606980715738, −14.449656792602319580640065395258, −13.45571468745627855314740471735, −12.61715033013226884284400125960, −11.64924730571221846740784395908, −10.77542637238643474675844682087, −10.023904191100210899258024503536, −9.23088711003691585540455438657, −7.3160284018044150186119432043, −6.2336770957235632449790145853, −5.21560058986919635135543155921, −4.39367333565820476653460720116, −2.7516121948685183791292683317, −1.77021476491544096396095554309, −0.2179329550281136987093590604, 1.248394223167636001905987997010, 3.3281276106536765070716394002, 4.83103847345474124041440258274, 5.56644800576888374704057409525, 6.10691046079612161906979627766, 7.48823506605915287447515106576, 8.41449564392946453048936540706, 9.77105020472539606084440069372, 10.53791205907772616181483986538, 12.134791691903262691718513082515, 12.85262017121130922358841347058, 13.57073859834098984992096683900, 14.82829152114738445069664822646, 15.87335994241196338332403782177, 16.64417012661301948024787840895, 17.32990025709300010061022861908, 18.05617790046217375667062142743, 19.006074123976112488798312552933, 20.85557630592304500522629640788, 21.42977251890905619991245275653, 22.2925758934615117426471683652, 23.24491469581197033162557544596, 23.90440084840806580828860147738, 24.783175474512239943857649604753, 25.48608849453527161696640421733

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