↑

Properties

Label 2-287-41.16-c1-0-2
Degree $2$
Conductor $287$
Sign $-0.952 + 0.303i$
Analytic cond. $2.29170$
Root an. cond. $1.51383$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.662 + 2.03i)2-s + 0.259·3-s + (−2.09 − 1.52i)4-s + (0.0549 + 0.0399i)5-s + (−0.171 + 0.528i)6-s + (0.309 + 0.951i)7-s + (1.03 − 0.750i)8-s − 2.93·9-s + (−0.117 + 0.0856i)10-s + (−1.64 + 1.19i)11-s + (−0.543 − 0.395i)12-s + (−2.03 + 6.26i)13-s − 2.14·14-s + (0.0142 + 0.0103i)15-s + (−0.758 − 2.33i)16-s + (0.263 − 0.191i)17-s + ⋯
L(s)  = 1  + (−0.468 + 1.44i)2-s + 0.149·3-s + (−1.04 − 0.762i)4-s + (0.0245 + 0.0178i)5-s + (−0.0700 + 0.215i)6-s + (0.116 + 0.359i)7-s + (0.365 − 0.265i)8-s − 0.977·9-s + (−0.0372 + 0.0270i)10-s + (−0.496 + 0.360i)11-s + (−0.157 − 0.114i)12-s + (−0.564 + 1.73i)13-s − 0.572·14-s + (0.00367 + 0.00267i)15-s + (−0.189 − 0.583i)16-s + (0.0639 − 0.0464i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.303i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.952 + 0.303i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (57, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.952 + 0.303i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.106916 - 0.688299i\)
\(L(\frac12)\) \(\approx\) \(0.106916 - 0.688299i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-0.309 - 0.951i)T \)
41 \( 1 + (4.22 + 4.80i)T \)
good2 \( 1 + (0.662 - 2.03i)T + (-1.61 - 1.17i)T^{2} \)
3 \( 1 - 0.259T + 3T^{2} \)
5 \( 1 + (-0.0549 - 0.0399i)T + (1.54 + 4.75i)T^{2} \)
11 \( 1 + (1.64 - 1.19i)T + (3.39 - 10.4i)T^{2} \)
13 \( 1 + (2.03 - 6.26i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-0.263 + 0.191i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-0.0335 - 0.103i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 + (1.89 - 5.83i)T + (-18.6 - 13.5i)T^{2} \)
29 \( 1 + (-6.81 - 4.94i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (-6.57 + 4.77i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-3.14 - 2.28i)T + (11.4 + 35.1i)T^{2} \)
43 \( 1 + (-3.05 + 9.41i)T + (-34.7 - 25.2i)T^{2} \)
47 \( 1 + (3.17 - 9.77i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-5.38 - 3.91i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (3.71 - 11.4i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (1.62 + 5.01i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + (-9.25 - 6.72i)T + (20.7 + 63.7i)T^{2} \)
71 \( 1 + (3.75 - 2.72i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + 1.16T + 73T^{2} \)
79 \( 1 - 2.46T + 79T^{2} \)
83 \( 1 + 1.53T + 83T^{2} \)
89 \( 1 + (-2.16 - 6.66i)T + (-72.0 + 52.3i)T^{2} \)
97 \( 1 + (5.68 + 4.12i)T + (29.9 + 92.2i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−12.13881772846951834305746279405, −11.56564109337622870077283588044, −10.03013000083703894856808656643, −9.143995518071733159263726510968, −8.417755856631721896419709356685, −7.50546868273581384385118476237, −6.54401075064391865165383738585, −5.63137305061747583858762177958, −4.53434517920688424400484916314, −2.50884013504455377030719994209, 0.57565889956387697552070938703, 2.55740253457590237360622943276, 3.29963788888022059259046313890, 4.90580658502210414724447354444, 6.22525695219385288372471418106, 8.006360586183215547064072448811, 8.460251694024046426113058487875, 9.822040234641687889715325936593, 10.37892504898011553537946408452, 11.19336526516418825034666769965

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