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Properties

Label 1-287-287.228-r0-0-0
Degree $1$
Conductor $287$
Sign $0.574 + 0.818i$
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  + (−0.104 − 0.994i)2-s + (0.5 + 0.866i)3-s + (−0.978 + 0.207i)4-s + (0.669 + 0.743i)5-s + (0.809 − 0.587i)6-s + (0.309 + 0.951i)8-s + (−0.5 + 0.866i)9-s + (0.669 − 0.743i)10-s + (−0.669 + 0.743i)11-s + (−0.669 − 0.743i)12-s + (0.809 − 0.587i)13-s + (−0.309 + 0.951i)15-s + (0.913 − 0.406i)16-s + (−0.669 + 0.743i)17-s + (0.913 + 0.406i)18-s + (−0.913 + 0.406i)19-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (0.5 + 0.866i)3-s + (−0.978 + 0.207i)4-s + (0.669 + 0.743i)5-s + (0.809 − 0.587i)6-s + (0.309 + 0.951i)8-s + (−0.5 + 0.866i)9-s + (0.669 − 0.743i)10-s + (−0.669 + 0.743i)11-s + (−0.669 − 0.743i)12-s + (0.809 − 0.587i)13-s + (−0.309 + 0.951i)15-s + (0.913 − 0.406i)16-s + (−0.669 + 0.743i)17-s + (0.913 + 0.406i)18-s + (−0.913 + 0.406i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.574 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.574 + 0.818i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (228, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.574 + 0.818i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.078789723 + 0.5603956626i\)
\(L(\frac12)\) \(\approx\) \(1.078789723 + 0.5603956626i\)
\(L(1)\) \(\approx\) \(1.086354074 + 0.1223880911i\)
\(L(1)\) \(\approx\) \(1.086354074 + 0.1223880911i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.104 - 0.994i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + (0.669 + 0.743i)T \)
11 \( 1 + (-0.669 + 0.743i)T \)
13 \( 1 + (0.809 - 0.587i)T \)
17 \( 1 + (-0.669 + 0.743i)T \)
19 \( 1 + (-0.913 + 0.406i)T \)
23 \( 1 + (-0.104 - 0.994i)T \)
29 \( 1 + (-0.309 + 0.951i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (0.669 + 0.743i)T \)
43 \( 1 + (-0.809 + 0.587i)T \)
47 \( 1 + (0.104 + 0.994i)T \)
53 \( 1 + (0.978 - 0.207i)T \)
59 \( 1 + (0.913 + 0.406i)T \)
61 \( 1 + (0.913 - 0.406i)T \)
67 \( 1 + (0.978 - 0.207i)T \)
71 \( 1 + (-0.309 - 0.951i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (0.5 - 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (-0.913 + 0.406i)T \)
97 \( 1 + (-0.309 + 0.951i)T \)
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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

−25.21054820311349840428614389383, −24.64990826686051451956949882369, −23.74414014897990882972164464001, −23.26912479508982272301385553262, −21.72844455538446755640273304893, −20.95595672148576198297229547470, −19.72723196536582822001698853393, −18.784318405582543427442357349225, −17.996927853621195784895811357328, −17.23810307037124910327205151542, −16.21583155001917571439315056589, −15.3424128451078438264431258486, −14.01222026462195112258225796151, −13.483622139768133500080687394180, −12.92204713453881031057267717927, −11.47504670608884661048701247601, −9.86455089189580872956331779008, −8.77245401587517274882121674164, −8.41185375875510583671507660949, −7.11231574098125816876023774918, −6.19445342410658688203750613454, −5.3185776917701961875096889989, −3.92860111931367875697451885683, −2.24409940068259839152456369947, −0.80438168647851812437793714282, 1.952389315794181945107230962289, 2.776993209610220686489571378623, 3.868409230954415517418754907587, 4.90941846428608741513284421058, 6.18465921666965894525417973637, 7.96011937991650972794509881058, 8.81238340633699081421966519212, 9.98615213205249005576995889287, 10.46777579895086994372656400518, 11.192388312651867257255098683866, 12.82062060665897106846300170929, 13.45195928213285318008988117497, 14.62162376416582997635181966877, 15.19208458374014598727690947248, 16.651245489298459756016562232623, 17.70093362750455561184107393445, 18.469550646028029186064719466587, 19.44488455477279490780167099018, 20.542948208724376020206886421061, 20.94561673578605437740696302623, 21.95924373744535499552155254776, 22.53683129334024068080520097665, 23.47184968457397661441277089393, 25.2172743515638651489358130250, 26.010239149000869854175781590956

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