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Properties

Label 2-286-11.9-c1-0-1
Degree $2$
Conductor $286$
Sign $0.319 - 0.947i$
Analytic cond. $2.28372$
Root an. cond. $1.51119$
Motivic weight $1$
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 + (−1.24 + 0.904i)3-s + (−0.809 − 0.587i)4-s + (0.590 + 1.81i)5-s + (0.475 + 1.46i)6-s + (−1.16 − 0.845i)7-s + (−0.809 + 0.587i)8-s + (−0.194 + 0.598i)9-s + 1.90·10-s + (−0.189 + 3.31i)11-s + 1.53·12-s + (−0.309 + 0.951i)13-s + (−1.16 + 0.845i)14-s + (−2.37 − 1.72i)15-s + (0.309 + 0.951i)16-s + (1.91 + 5.89i)17-s + ⋯
L(s)  = 1  + (0.218 − 0.672i)2-s + (−0.719 + 0.522i)3-s + (−0.404 − 0.293i)4-s + (0.263 + 0.812i)5-s + (0.194 + 0.597i)6-s + (−0.439 − 0.319i)7-s + (−0.286 + 0.207i)8-s + (−0.0648 + 0.199i)9-s + 0.603·10-s + (−0.0571 + 0.998i)11-s + 0.444·12-s + (−0.0857 + 0.263i)13-s + (−0.310 + 0.225i)14-s + (−0.614 − 0.446i)15-s + (0.0772 + 0.237i)16-s + (0.464 + 1.43i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 286 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 286 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.319 - 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(286\)    =    \(2 \cdot 11 \cdot 13\)
Sign: $0.319 - 0.947i$
Analytic conductor: \(2.28372\)
Root analytic conductor: \(1.51119\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{286} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 286,\ (\ :1/2),\ 0.319 - 0.947i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.714912 + 0.513187i\)
\(L(\frac12)\) \(\approx\) \(0.714912 + 0.513187i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.309 + 0.951i)T \)
11 \( 1 + (0.189 - 3.31i)T \)
13 \( 1 + (0.309 - 0.951i)T \)
good3 \( 1 + (1.24 - 0.904i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.590 - 1.81i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (1.16 + 0.845i)T + (2.16 + 6.65i)T^{2} \)
17 \( 1 + (-1.91 - 5.89i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (3.17 - 2.30i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 2.73T + 23T^{2} \)
29 \( 1 + (1.72 + 1.25i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-0.0478 + 0.147i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-4.15 - 3.01i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (1.88 - 1.36i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 8.46T + 43T^{2} \)
47 \( 1 + (-8.42 + 6.11i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-1.26 + 3.89i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (8.56 + 6.22i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-3.92 - 12.0i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 1.06T + 67T^{2} \)
71 \( 1 + (1.25 + 3.87i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-6.19 - 4.50i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (-1.78 + 5.50i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-5.48 - 16.8i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 - 8.95T + 89T^{2} \)
97 \( 1 + (1.22 - 3.78i)T + (-78.4 - 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.89602456473725605838275120273, −10.86177263313410397154653875412, −10.31675037393163648811793829575, −9.768955302202710699534942999726, −8.266214220192440021530812432667, −6.85773374616639059137162627241, −5.92662763600305037093969193043, −4.72705598504865991618343733423, −3.66166437092516942367043403911, −2.12461957394693876659717885051, 0.67775137034056088898120899047, 3.13628126932830571148176971772, 4.89231279878400445247575211510, 5.69226144285083531624744125556, 6.50996148842161095312414693733, 7.55481435533235850428642996934, 8.855392888200110698878308643123, 9.354565898620922000288337619764, 10.91669583013073036963919989927, 11.86928395532872102583181516621

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