↑

Properties

Label 2-287-41.18-c1-0-9
Degree $2$
Conductor $287$
Sign $-0.268 - 0.963i$
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  + (0.718 + 2.21i)2-s + 0.707·3-s + (−2.75 + 2.00i)4-s + (3.19 − 2.32i)5-s + (0.508 + 1.56i)6-s + (−0.309 + 0.951i)7-s + (−2.64 − 1.91i)8-s − 2.49·9-s + (7.43 + 5.40i)10-s + (0.354 + 0.257i)11-s + (−1.94 + 1.41i)12-s + (1.94 + 5.97i)13-s − 2.32·14-s + (2.26 − 1.64i)15-s + (0.242 − 0.745i)16-s + (−2.54 − 1.84i)17-s + ⋯
L(s)  = 1  + (0.507 + 1.56i)2-s + 0.408·3-s + (−1.37 + 1.00i)4-s + (1.43 − 1.03i)5-s + (0.207 + 0.638i)6-s + (−0.116 + 0.359i)7-s + (−0.934 − 0.678i)8-s − 0.833·9-s + (2.35 + 1.70i)10-s + (0.106 + 0.0775i)11-s + (−0.562 + 0.408i)12-s + (0.538 + 1.65i)13-s − 0.621·14-s + (0.584 − 0.424i)15-s + (0.0605 − 0.186i)16-s + (−0.617 − 0.448i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.268 - 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.268 - 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.268 - 0.963i$
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: \((2,\ 287,\ (\ :1/2),\ -0.268 - 0.963i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23130 + 1.62159i\)
\(L(\frac12)\) \(\approx\) \(1.23130 + 1.62159i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-6.33 - 0.898i)T \)
good2 \( 1 + (-0.718 - 2.21i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 - 0.707T + 3T^{2} \)
5 \( 1 + (-3.19 + 2.32i)T + (1.54 - 4.75i)T^{2} \)
11 \( 1 + (-0.354 - 0.257i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (-1.94 - 5.97i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.54 + 1.84i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.48 + 4.57i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.66 + 5.12i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (-1.48 + 1.08i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.97 + 1.43i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.74 - 1.99i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (0.457 + 1.40i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (3.34 + 10.2i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (4.30 - 3.13i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (1.37 + 4.22i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (3.61 - 11.1i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (-0.599 + 0.435i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-12.0 - 8.77i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 - 6.87T + 73T^{2} \)
79 \( 1 + 6.98T + 79T^{2} \)
83 \( 1 + 8.01T + 83T^{2} \)
89 \( 1 + (4.77 - 14.6i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (-1.19 + 0.870i)T + (29.9 - 92.2i)T^{2} \)
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

−12.55263354152204676599621797798, −11.32009426510305667260473584502, −9.562711417057919367997759264827, −8.893450114630411045114754131015, −8.477649356250451887543977149911, −6.85333860722321737700232466179, −6.15621717825454799308161156583, −5.24730780589400219728015412503, −4.36127375638609557663937551217, −2.23901902691537098215795459187, 1.71324264933273748573397503774, 2.92759819477174128703628129527, 3.53253083466048375316969554045, 5.42456329952485296568296196116, 6.16467328613218113887708606626, 7.82145934235241582512635648092, 9.229168653161275267934089499700, 10.00623071922277586592850500980, 10.71248090144076329005477747038, 11.23777366813467646938761514737

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