↑

Properties

Label 2-287-287.9-c1-0-23
Degree $2$
Conductor $287$
Sign $0.0376 + 0.999i$
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  + (1.12 + 0.648i)2-s + (0.752 − 2.80i)3-s + (−0.159 − 0.276i)4-s + (−1.50 − 0.867i)5-s + (2.66 − 2.66i)6-s + (−0.701 + 2.55i)7-s − 3.00i·8-s + (−4.72 − 2.72i)9-s + (−1.12 − 1.94i)10-s + (0.603 − 2.25i)11-s + (−0.895 + 0.239i)12-s + (3.55 + 3.55i)13-s + (−2.44 + 2.40i)14-s + (−3.56 + 3.56i)15-s + (1.63 − 2.82i)16-s + (1.30 + 0.348i)17-s + ⋯
L(s)  = 1  + (0.794 + 0.458i)2-s + (0.434 − 1.62i)3-s + (−0.0796 − 0.138i)4-s + (−0.672 − 0.388i)5-s + (1.08 − 1.08i)6-s + (−0.265 + 0.964i)7-s − 1.06i·8-s + (−1.57 − 0.909i)9-s + (−0.355 − 0.616i)10-s + (0.181 − 0.678i)11-s + (−0.258 + 0.0692i)12-s + (0.986 + 0.986i)13-s + (−0.652 + 0.644i)14-s + (−0.921 + 0.921i)15-s + (0.407 − 0.706i)16-s + (0.315 + 0.0846i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0376 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0376 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.0376 + 0.999i$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 0.0376 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.29308 - 1.24528i\)
\(L(\frac12)\) \(\approx\) \(1.29308 - 1.24528i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + (0.701 - 2.55i)T \)
41 \( 1 + (-5.82 + 2.66i)T \)
good2 \( 1 + (-1.12 - 0.648i)T + (1 + 1.73i)T^{2} \)
3 \( 1 + (-0.752 + 2.80i)T + (-2.59 - 1.5i)T^{2} \)
5 \( 1 + (1.50 + 0.867i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.603 + 2.25i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.55 - 3.55i)T + 13iT^{2} \)
17 \( 1 + (-1.30 - 0.348i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.586 - 2.18i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-3.47 + 6.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.651 - 0.651i)T + 29iT^{2} \)
31 \( 1 + (-2.55 - 4.42i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.37 + 5.84i)T + (-18.5 - 32.0i)T^{2} \)
43 \( 1 - 11.9iT - 43T^{2} \)
47 \( 1 + (-2.86 - 10.6i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (2.38 - 8.89i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (0.182 + 0.316i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.12 + 0.649i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (15.2 + 4.07i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (2.32 + 2.32i)T + 71iT^{2} \)
73 \( 1 + (4.26 - 2.46i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.101 + 0.0273i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 - 6.06T + 83T^{2} \)
89 \( 1 + (-2.82 + 0.756i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (0.441 - 0.441i)T - 97iT^{2} \)
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

−12.22349111598761513368468027273, −11.05109132379448579041673192069, −9.201347972651197969585365839323, −8.563297667965728724543098705847, −7.57502720770578935078889089866, −6.31613093935045565418893537816, −6.04847502826320633753389584701, −4.39684325381887555450311186104, −2.97755826727822412509614654453, −1.16319071117190837611555788981, 3.09997131734343058672034565635, 3.68051256823532809428381131744, 4.43016371766714331387389417477, 5.49761816545111661127423510242, 7.36250844055927643317458551825, 8.315335360926940018241614420155, 9.434116312184302118741464307407, 10.36059464831582951473639417156, 11.08100556441057019133386532440, 11.80625654384061965009413017101

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