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Properties

Label 1-287-287.10-r1-0-0
Degree $1$
Conductor $287$
Sign $-0.357 - 0.934i$
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  + (−0.978 − 0.207i)2-s + (0.5 + 0.866i)3-s + (0.913 + 0.406i)4-s + (0.104 + 0.994i)5-s + (−0.309 − 0.951i)6-s + (−0.809 − 0.587i)8-s + (−0.5 + 0.866i)9-s + (0.104 − 0.994i)10-s + (−0.104 + 0.994i)11-s + (0.104 + 0.994i)12-s + (−0.309 − 0.951i)13-s + (−0.809 + 0.587i)15-s + (0.669 + 0.743i)16-s + (0.104 − 0.994i)17-s + (0.669 − 0.743i)18-s + (−0.669 − 0.743i)19-s + ⋯
L(s)  = 1  + (−0.978 − 0.207i)2-s + (0.5 + 0.866i)3-s + (0.913 + 0.406i)4-s + (0.104 + 0.994i)5-s + (−0.309 − 0.951i)6-s + (−0.809 − 0.587i)8-s + (−0.5 + 0.866i)9-s + (0.104 − 0.994i)10-s + (−0.104 + 0.994i)11-s + (0.104 + 0.994i)12-s + (−0.309 − 0.951i)13-s + (−0.809 + 0.587i)15-s + (0.669 + 0.743i)16-s + (0.104 − 0.994i)17-s + (0.669 − 0.743i)18-s + (−0.669 − 0.743i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.357 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.357 - 0.934i$
Analytic conductor: \(30.8424\)
Root analytic conductor: \(30.8424\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (1:\ ),\ -0.357 - 0.934i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.01281038068 + 0.01861073381i\)
\(L(\frac12)\) \(\approx\) \(0.01281038068 + 0.01861073381i\)
\(L(1)\) \(\approx\) \(0.5988794781 + 0.2388490730i\)
\(L(1)\) \(\approx\) \(0.5988794781 + 0.2388490730i\)

Euler product

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

−24.69364505298773002415916606886, −24.024444719538885989118536862021, −23.574488428649783869871635702817, −21.57500243766207221261752133583, −20.80414866009418555911335869723, −19.84061692137836316813198203350, −19.155648443716780801389959142809, −18.47691753203590583642946635351, −17.2190103224261413899971344271, −16.77016474395584181828347818442, −15.658950404263997570865929093628, −14.47477323360015349970038859820, −13.56662453396434213841359661612, −12.37302486264389835908303896796, −11.68548239813229465701962845829, −10.29351215002047719343265674654, −9.10508298210554352722538509337, −8.4742800770652132611055043395, −7.74409729368425803428997000385, −6.445774735637728524704293160968, −5.67201928909578168357747006683, −3.78894354628427873130004826835, −2.15160115236927459250796562565, −1.33271300769603083802538128690, −0.00847268214610709802135945544, 2.22482163239039598388879602875, 2.868807249054998306410599650280, 4.15471672748691201327651849853, 5.72542567408290155587547562756, 7.170623079728822501793542210056, 7.82944865971105508328051145511, 9.15103859303898582567459043826, 9.91553555958162714994311941409, 10.607578605134194033320844732380, 11.46211874427523746354897231898, 12.79453128403445598784961315120, 14.2290049344914382290023210822, 15.177062916848318308352518249692, 15.66410586444103535298983625422, 16.89724603731858636916929087256, 17.82282114749521279810330745379, 18.58147706587351320938762466411, 19.76208381098397957595354018454, 20.25524002506152330329051610965, 21.25272408438118567570763411959, 22.167911273373945392979385072, 22.907307314478182001797769220253, 24.554855673634876084453200632331, 25.5340176611559750176834550105, 25.99335814448787543692067019472

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