Properties

Label 2-287-41.18-c1-0-11
Degree $2$
Conductor $287$
Sign $-0.325 + 0.945i$
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  + (0.228 + 0.703i)2-s − 2.48·3-s + (1.17 − 0.854i)4-s + (−2.55 + 1.85i)5-s + (−0.568 − 1.75i)6-s + (0.309 − 0.951i)7-s + (2.06 + 1.50i)8-s + 3.19·9-s + (−1.88 − 1.37i)10-s + (−3.23 − 2.35i)11-s + (−2.92 + 2.12i)12-s + (−1.58 − 4.88i)13-s + 0.739·14-s + (6.36 − 4.62i)15-s + (0.314 − 0.969i)16-s + (−4.58 − 3.33i)17-s + ⋯
L(s)  = 1  + (0.161 + 0.497i)2-s − 1.43·3-s + (0.587 − 0.427i)4-s + (−1.14 + 0.830i)5-s + (−0.232 − 0.714i)6-s + (0.116 − 0.359i)7-s + (0.730 + 0.530i)8-s + 1.06·9-s + (−0.597 − 0.434i)10-s + (−0.975 − 0.709i)11-s + (−0.844 + 0.613i)12-s + (−0.440 − 1.35i)13-s + 0.197·14-s + (1.64 − 1.19i)15-s + (0.0787 − 0.242i)16-s + (−1.11 − 0.807i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.325 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.325 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.325 + 0.945i$
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: \((2,\ 287,\ (\ :1/2),\ -0.325 + 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.185414 - 0.260027i\)
\(L(\frac12)\) \(\approx\) \(0.185414 - 0.260027i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-4.70 - 4.33i)T \)
good2 \( 1 + (-0.228 - 0.703i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + 2.48T + 3T^{2} \)
5 \( 1 + (2.55 - 1.85i)T + (1.54 - 4.75i)T^{2} \)
11 \( 1 + (3.23 + 2.35i)T + (3.39 + 10.4i)T^{2} \)
13 \( 1 + (1.58 + 4.88i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (4.58 + 3.33i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.188 - 0.579i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + (1.65 + 5.08i)T + (-18.6 + 13.5i)T^{2} \)
29 \( 1 + (7.91 - 5.75i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-5.18 - 3.76i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (2.83 - 2.06i)T + (11.4 - 35.1i)T^{2} \)
43 \( 1 + (1.16 + 3.59i)T + (-34.7 + 25.2i)T^{2} \)
47 \( 1 + (2.32 + 7.16i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (6.29 - 4.57i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.413 - 1.27i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-2.33 + 7.17i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + (7.11 - 5.17i)T + (20.7 - 63.7i)T^{2} \)
71 \( 1 + (-3.25 - 2.36i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + 13.6T + 73T^{2} \)
79 \( 1 - 4.59T + 79T^{2} \)
83 \( 1 - 11.0T + 83T^{2} \)
89 \( 1 + (-2.23 + 6.87i)T + (-72.0 - 52.3i)T^{2} \)
97 \( 1 + (5.42 - 3.94i)T + (29.9 - 92.2i)T^{2} \)
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

−11.29783697281670161947044616491, −10.75654809613005740129164752112, −10.35681784325480130044084015121, −8.178771044782946371079631065345, −7.31111668638312933620819158293, −6.61242980697807199223327137263, −5.55271241913749946505033877569, −4.73435769486745691922435289302, −2.94589413716086108057186206061, −0.25254390927774214905683516314, 2.00758395280424046449160044631, 4.11471632282402386077304655431, 4.73030583953038808865784478149, 6.08933009996092779221711012170, 7.24161080918458615327766723326, 8.031827303585930543615467560675, 9.454811619957480879021059218323, 10.74071800558344332528799527284, 11.49043204591634049359148944159, 11.85571994438407524935482243107

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