↑

Properties

Label 2-287-287.286-c2-0-1
Degree $2$
Conductor $287$
Sign $-0.483 - 0.875i$
Analytic cond. $7.82018$
Root an. cond. $2.79645$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.16·2-s − 4.43·3-s − 2.63·4-s − 9.05i·5-s − 5.17·6-s + (6.69 − 2.05i)7-s − 7.74·8-s + 10.6·9-s − 10.5i·10-s + 15.2i·11-s + 11.6·12-s − 11.9·13-s + (7.80 − 2.40i)14-s + 40.1i·15-s + 1.50·16-s − 17.3·17-s + ⋯
L(s)  = 1  + 0.583·2-s − 1.47·3-s − 0.659·4-s − 1.81i·5-s − 0.862·6-s + (0.955 − 0.294i)7-s − 0.968·8-s + 1.18·9-s − 1.05i·10-s + 1.38i·11-s + 0.974·12-s − 0.918·13-s + (0.557 − 0.171i)14-s + 2.67i·15-s + 0.0942·16-s − 1.01·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.483 - 0.875i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.483 - 0.875i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.483 - 0.875i$
Analytic conductor: \(7.82018\)
Root analytic conductor: \(2.79645\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (286, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1),\ -0.483 - 0.875i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.0623123 + 0.105556i\)
\(L(\frac12)\) \(\approx\) \(0.0623123 + 0.105556i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (-6.69 + 2.05i)T \)
41 \( 1 + (-8.37 - 40.1i)T \)
good2 \( 1 - 1.16T + 4T^{2} \)
3 \( 1 + 4.43T + 9T^{2} \)
5 \( 1 + 9.05iT - 25T^{2} \)
11 \( 1 - 15.2iT - 121T^{2} \)
13 \( 1 + 11.9T + 169T^{2} \)
17 \( 1 + 17.3T + 289T^{2} \)
19 \( 1 - 0.333T + 361T^{2} \)
23 \( 1 - 14.2T + 529T^{2} \)
29 \( 1 - 9.15iT - 841T^{2} \)
31 \( 1 - 50.9iT - 961T^{2} \)
37 \( 1 - 16.3T + 1.36e3T^{2} \)
43 \( 1 + 38.9T + 1.84e3T^{2} \)
47 \( 1 + 79.8T + 2.20e3T^{2} \)
53 \( 1 - 28.3iT - 2.80e3T^{2} \)
59 \( 1 + 89.3iT - 3.48e3T^{2} \)
61 \( 1 - 35.2iT - 3.72e3T^{2} \)
67 \( 1 + 62.8iT - 4.48e3T^{2} \)
71 \( 1 + 30.5iT - 5.04e3T^{2} \)
73 \( 1 - 19.9iT - 5.32e3T^{2} \)
79 \( 1 - 38.5iT - 6.24e3T^{2} \)
83 \( 1 + 113. iT - 6.88e3T^{2} \)
89 \( 1 + 141.T + 7.92e3T^{2} \)
97 \( 1 + 86.9T + 9.40e3T^{2} \)
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

−12.16988350652055667949175039059, −11.33712452587682608272231633586, −10.02725766805830846080087975964, −9.143090095046279053035353996004, −8.116032350425269496004716820792, −6.74395159462937334266017758247, −5.26294197010909336345003363596, −4.82377219166961741192923628706, −4.48095643626834283116790797491, −1.41141886995284293816726764371, 0.06492614549385974048246213492, 2.72633012744420030240778355634, 4.17923448534367564549610309811, 5.35176056098238135800924905478, 6.05577119183415205294112341757, 6.95735818507225145089831051973, 8.264028248384200652402107682589, 9.708695321845000958557000073245, 10.80395848467126384335572741487, 11.33867123211566921328357559116

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