↑

Properties

Label 2-287-41.18-c1-0-1
Degree $2$
Conductor $287$
Sign $-0.0714 + 0.997i$
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  + (0.805 + 2.47i)2-s − 2.12·3-s + (−3.87 + 2.81i)4-s + (−0.434 + 0.315i)5-s + (−1.71 − 5.26i)6-s + (−0.309 + 0.951i)7-s + (−5.89 − 4.28i)8-s + 1.51·9-s + (−1.13 − 0.822i)10-s + (1.37 + 0.995i)11-s + (8.24 − 5.98i)12-s + (−1.30 − 4.01i)13-s − 2.60·14-s + (0.922 − 0.670i)15-s + (2.90 − 8.93i)16-s + (−2.11 − 1.53i)17-s + ⋯
L(s)  = 1  + (0.569 + 1.75i)2-s − 1.22·3-s + (−1.93 + 1.40i)4-s + (−0.194 + 0.141i)5-s + (−0.698 − 2.15i)6-s + (−0.116 + 0.359i)7-s + (−2.08 − 1.51i)8-s + 0.505·9-s + (−0.357 − 0.260i)10-s + (0.413 + 0.300i)11-s + (2.37 − 1.72i)12-s + (−0.361 − 1.11i)13-s − 0.696·14-s + (0.238 − 0.173i)15-s + (0.725 − 2.23i)16-s + (−0.512 − 0.372i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0714 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0714 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.0714 + 0.997i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (141, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ -0.0714 + 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.305649 - 0.328311i\)
\(L(\frac12)\) \(\approx\) \(0.305649 - 0.328311i\)
\(L(\frac{3}{2})\) 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 \( 1 + (5.66 + 2.98i)T \)
good2 \( 1 + (-0.805 - 2.47i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + 2.12T + 3T^{2} \)
5 \( 1 + (0.434 - 0.315i)T + (1.54 - 4.75i)T^{2} \)
11 \( 1 + (-1.37 - 0.995i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (1.30 + 4.01i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.11 + 1.53i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.882 - 2.71i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (0.0388 + 0.119i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (7.17 - 5.21i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (0.477 + 0.347i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (7.23 - 5.25i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (-1.82 - 5.61i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (-0.726 - 2.23i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-8.02 + 5.83i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.17 + 3.60i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (3.97 - 12.2i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-7.57 + 5.50i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-2.45 - 1.78i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 - 8.72T + 73T^{2} \)
79 \( 1 - 17.3T + 79T^{2} \)
83 \( 1 + 14.1T + 83T^{2} \)
89 \( 1 + (3.39 - 10.4i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-14.4 + 10.4i)T + (29.9 - 92.2i)T^{2} \)
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

−12.61202580507271544054332958996, −11.90041718995130945871544111557, −10.73278514658555065854718698242, −9.422392421011888019955444116472, −8.325004888178673476398633712666, −7.27111909078145033815691196504, −6.52434871714401933347042138426, −5.52549795092286937831405727791, −5.04685763024728517039381320747, −3.61676979198423303221408680470, 0.32049976815817258338838719018, 2.04727057788386228589906178888, 3.81490428227371296514682911764, 4.60505812581933998707604007781, 5.68041718201512413876561363727, 6.81344313074244311000131881779, 8.744319855302264420523100819024, 9.682360783983047717395283385350, 10.68244414503764897480867200954, 11.25804932799886231240909756461

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