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Properties

Label 1-287-287.110-r0-0-0
Degree $1$
Conductor $287$
Sign $0.453 - 0.891i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.406 − 0.913i)2-s + (0.965 − 0.258i)3-s + (−0.669 − 0.743i)4-s + (0.207 + 0.978i)5-s + (0.156 − 0.987i)6-s + (−0.951 + 0.309i)8-s + (0.866 − 0.5i)9-s + (0.978 + 0.207i)10-s + (0.544 + 0.838i)11-s + (−0.838 − 0.544i)12-s + (0.987 + 0.156i)13-s + (0.453 + 0.891i)15-s + (−0.104 + 0.994i)16-s + (0.838 − 0.544i)17-s + (−0.104 − 0.994i)18-s + (−0.629 − 0.777i)19-s + ⋯
L(s)  = 1  + (0.406 − 0.913i)2-s + (0.965 − 0.258i)3-s + (−0.669 − 0.743i)4-s + (0.207 + 0.978i)5-s + (0.156 − 0.987i)6-s + (−0.951 + 0.309i)8-s + (0.866 − 0.5i)9-s + (0.978 + 0.207i)10-s + (0.544 + 0.838i)11-s + (−0.838 − 0.544i)12-s + (0.987 + 0.156i)13-s + (0.453 + 0.891i)15-s + (−0.104 + 0.994i)16-s + (0.838 − 0.544i)17-s + (−0.104 − 0.994i)18-s + (−0.629 − 0.777i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.453 - 0.891i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (110, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.453 - 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.786476170 - 1.095914915i\)
\(L(\frac12)\) \(\approx\) \(1.786476170 - 1.095914915i\)
\(L(1)\) \(\approx\) \(1.546495544 - 0.7136813194i\)
\(L(1)\) \(\approx\) \(1.546495544 - 0.7136813194i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.406 - 0.913i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 + (0.207 + 0.978i)T \)
11 \( 1 + (0.544 + 0.838i)T \)
13 \( 1 + (0.987 + 0.156i)T \)
17 \( 1 + (0.838 - 0.544i)T \)
19 \( 1 + (-0.629 - 0.777i)T \)
23 \( 1 + (-0.913 - 0.406i)T \)
29 \( 1 + (0.891 - 0.453i)T \)
31 \( 1 + (-0.978 - 0.207i)T \)
37 \( 1 + (-0.978 + 0.207i)T \)
43 \( 1 + (0.587 + 0.809i)T \)
47 \( 1 + (-0.933 + 0.358i)T \)
53 \( 1 + (0.0523 - 0.998i)T \)
59 \( 1 + (0.104 + 0.994i)T \)
61 \( 1 + (-0.994 - 0.104i)T \)
67 \( 1 + (0.998 + 0.0523i)T \)
71 \( 1 + (-0.453 + 0.891i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (-0.258 + 0.965i)T \)
83 \( 1 - T \)
89 \( 1 + (0.777 - 0.629i)T \)
97 \( 1 + (-0.453 - 0.891i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−25.50576512282694991977722334906, −24.92582688360301401927287124136, −24.04130034779206697693090628154, −23.30840945008583539134624330335, −21.85732328873070874917120344168, −21.29908900427464010071334436333, −20.48419126730470370473298997628, −19.36025267428451679141772711720, −18.356963028222801669439596990150, −17.10479288154800656502427979703, −16.27472743234623719331283308664, −15.704036717895873110152564431916, −14.437795251828910688741956066934, −13.88317039386260547238672879320, −12.97090492737357433598615780170, −12.12240203783994884594961925885, −10.356915851918671049566584177853, −9.11079497849984828197143504163, −8.5103747061347208168799729222, −7.80587568075219969113699031265, −6.28185192892991813727453394758, −5.37388895833170628437298411273, −4.0466591253272215557775791234, −3.44315130812804988289671232005, −1.52889220512988031296907661232, 1.53165474716435098994771381162, 2.510715429722892274945560437792, 3.4767024487326460137573089805, 4.413690172581413954472679568351, 6.10384841236270739760381867518, 7.06938482428644248353580142381, 8.43472212928120932295377100559, 9.52241810628170001232835822632, 10.22848610069532786828323306265, 11.35469325285871253222137572738, 12.37927319016292330898026114165, 13.39486050141176786793205998086, 14.206410242171120319112047078864, 14.76360251035201532105291186311, 15.77036512667044775974738520638, 17.69291551093108744145385235906, 18.35126357965335235205577207717, 19.170084548990950540977336188, 19.926123033391093644182031745976, 20.861806753916041469223120227170, 21.55452289610369700262140926053, 22.622457289641512361404301888845, 23.32777303899757868014728069502, 24.40397945238278254636195639901, 25.63616390603130784809030259116

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