↑

Properties

Label 1-287-287.138-r0-0-0
Degree $1$
Conductor $287$
Sign $0.909 - 0.415i$
Analytic cond. $1.33282$
Root an. cond. $1.33282$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.207 − 0.978i)2-s + (−0.258 − 0.965i)3-s + (−0.913 + 0.406i)4-s + (−0.994 − 0.104i)5-s + (−0.891 + 0.453i)6-s + (0.587 + 0.809i)8-s + (−0.866 + 0.5i)9-s + (0.104 + 0.994i)10-s + (0.777 + 0.629i)11-s + (0.629 + 0.777i)12-s + (0.453 + 0.891i)13-s + (0.156 + 0.987i)15-s + (0.669 − 0.743i)16-s + (−0.629 + 0.777i)17-s + (0.669 + 0.743i)18-s + (−0.998 + 0.0523i)19-s + ⋯
L(s)  = 1  + (−0.207 − 0.978i)2-s + (−0.258 − 0.965i)3-s + (−0.913 + 0.406i)4-s + (−0.994 − 0.104i)5-s + (−0.891 + 0.453i)6-s + (0.587 + 0.809i)8-s + (−0.866 + 0.5i)9-s + (0.104 + 0.994i)10-s + (0.777 + 0.629i)11-s + (0.629 + 0.777i)12-s + (0.453 + 0.891i)13-s + (0.156 + 0.987i)15-s + (0.669 − 0.743i)16-s + (−0.629 + 0.777i)17-s + (0.669 + 0.743i)18-s + (−0.998 + 0.0523i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.415i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.909 - 0.415i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (138, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.909 - 0.415i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5725664308 - 0.1247142504i\)
\(L(\frac12)\) \(\approx\) \(0.5725664308 - 0.1247142504i\)
\(L(1)\) \(\approx\) \(0.5709018427 - 0.3106959935i\)
\(L(1)\) \(\approx\) \(0.5709018427 - 0.3106959935i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.207 - 0.978i)T \)
3 \( 1 + (-0.258 - 0.965i)T \)
5 \( 1 + (-0.994 - 0.104i)T \)
11 \( 1 + (0.777 + 0.629i)T \)
13 \( 1 + (0.453 + 0.891i)T \)
17 \( 1 + (-0.629 + 0.777i)T \)
19 \( 1 + (-0.998 + 0.0523i)T \)
23 \( 1 + (0.978 - 0.207i)T \)
29 \( 1 + (-0.987 + 0.156i)T \)
31 \( 1 + (-0.104 - 0.994i)T \)
37 \( 1 + (-0.104 + 0.994i)T \)
43 \( 1 + (0.951 + 0.309i)T \)
47 \( 1 + (0.838 + 0.544i)T \)
53 \( 1 + (-0.358 + 0.933i)T \)
59 \( 1 + (-0.669 - 0.743i)T \)
61 \( 1 + (-0.743 - 0.669i)T \)
67 \( 1 + (0.933 + 0.358i)T \)
71 \( 1 + (-0.156 + 0.987i)T \)
73 \( 1 + (0.866 + 0.5i)T \)
79 \( 1 + (0.965 + 0.258i)T \)
83 \( 1 - T \)
89 \( 1 + (0.0523 + 0.998i)T \)
97 \( 1 + (-0.156 - 0.987i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.66500269746962921378775557482, −24.75012386832511299590843304469, −23.71020303318950886982935541651, −22.83488963733053454200959739586, −22.41731391430076033757577007439, −21.234741605320906475073834463585, −19.99999605036791943508743024472, −19.21672953442633670850809811206, −18.06640965105876126725114052861, −17.05941069723509051390010870437, −16.3151941086837061334699942427, −15.48927657355904714271861324383, −14.94917979199625214687624736829, −13.917340921383576789195896896140, −12.58901897102462465769338042366, −11.2128157729381964277638481469, −10.60248791372874838358760246964, −9.11738095041603213360135226244, −8.648084121701230683044812092465, −7.38979627077671553786596203112, −6.29225114651034989309473491142, −5.21542328657386590534562450841, −4.18257182488784964938500662201, −3.33933349159317170883223550763, −0.514516571823492908954417044347, 1.216349158028796040237384068123, 2.28270210161892779367967645717, 3.791272989830994527687241109282, 4.63437648903279891709888210922, 6.3483764998358288608307678863, 7.409591854683783359498823082171, 8.45129511173132440077688963688, 9.22311232097333182727870430418, 10.94388527279088301522433266338, 11.35831877937423056986737169850, 12.42298075825878536239418930507, 12.9158406015365009370768525471, 14.1221455730775391695957789873, 15.158117345366519046151532908, 16.85177416002430536317344105187, 17.23481959059734597248599681992, 18.66020774426072973667245692954, 19.0135697707735749360370199324, 19.87767220187544067708325038712, 20.63680308679282846774746036853, 21.9638908083438999679733953308, 22.80437950744213464441772693393, 23.52266828473852494683064439185, 24.30043376930238052553914277332, 25.6174895265702200238304803886

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