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Properties

Label 1-287-287.107-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: $\chi_{287} (107, \cdot )$
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

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

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