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Properties

Label 1-287-287.75-r0-0-0
Degree $1$
Conductor $287$
Sign $0.900 - 0.434i$
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  + (0.994 + 0.104i)2-s + (0.965 − 0.258i)3-s + (0.978 + 0.207i)4-s + (−0.743 − 0.669i)5-s + (0.987 − 0.156i)6-s + (0.951 + 0.309i)8-s + (0.866 − 0.5i)9-s + (−0.669 − 0.743i)10-s + (0.0523 − 0.998i)11-s + (0.998 − 0.0523i)12-s + (0.156 + 0.987i)13-s + (−0.891 − 0.453i)15-s + (0.913 + 0.406i)16-s + (−0.998 − 0.0523i)17-s + (0.913 − 0.406i)18-s + (−0.933 + 0.358i)19-s + ⋯
L(s)  = 1  + (0.994 + 0.104i)2-s + (0.965 − 0.258i)3-s + (0.978 + 0.207i)4-s + (−0.743 − 0.669i)5-s + (0.987 − 0.156i)6-s + (0.951 + 0.309i)8-s + (0.866 − 0.5i)9-s + (−0.669 − 0.743i)10-s + (0.0523 − 0.998i)11-s + (0.998 − 0.0523i)12-s + (0.156 + 0.987i)13-s + (−0.891 − 0.453i)15-s + (0.913 + 0.406i)16-s + (−0.998 − 0.0523i)17-s + (0.913 − 0.406i)18-s + (−0.933 + 0.358i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.900 - 0.434i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $0.900 - 0.434i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (75, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ 0.900 - 0.434i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.737293675 - 0.6251011513i\)
\(L(\frac12)\) \(\approx\) \(2.737293675 - 0.6251011513i\)
\(L(1)\) \(\approx\) \(2.209875793 - 0.2867160047i\)
\(L(1)\) \(\approx\) \(2.209875793 - 0.2867160047i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (0.994 + 0.104i)T \)
3 \( 1 + (0.965 - 0.258i)T \)
5 \( 1 + (-0.743 - 0.669i)T \)
11 \( 1 + (0.0523 - 0.998i)T \)
13 \( 1 + (0.156 + 0.987i)T \)
17 \( 1 + (-0.998 - 0.0523i)T \)
19 \( 1 + (-0.933 + 0.358i)T \)
23 \( 1 + (0.104 - 0.994i)T \)
29 \( 1 + (-0.453 + 0.891i)T \)
31 \( 1 + (0.669 + 0.743i)T \)
37 \( 1 + (0.669 - 0.743i)T \)
43 \( 1 + (-0.587 + 0.809i)T \)
47 \( 1 + (-0.629 - 0.777i)T \)
53 \( 1 + (0.544 + 0.838i)T \)
59 \( 1 + (-0.913 + 0.406i)T \)
61 \( 1 + (-0.406 + 0.913i)T \)
67 \( 1 + (-0.838 + 0.544i)T \)
71 \( 1 + (0.891 - 0.453i)T \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (-0.258 + 0.965i)T \)
83 \( 1 - T \)
89 \( 1 + (-0.358 - 0.933i)T \)
97 \( 1 + (0.891 + 0.453i)T \)
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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

−25.57763339565980076064428464718, −24.68087380615618303261983685025, −23.68418394195364268852201399482, −22.778023262732720771571869622283, −22.09371637427213327408289017946, −21.06213707192202798952184031619, −20.114543986506470199116729679757, −19.67332837374904176707785809417, −18.66205861702241076410655820781, −17.27859702225407654456474159731, −15.65921161977423049758486097481, −15.312429903471033615133623053564, −14.72239255969435480263113977964, −13.49695412411857827574173621429, −12.85210062028917019800951269460, −11.61282873055875934231546405030, −10.66746383652493975153989959380, −9.7439383767131795983777881965, −8.16465163560512190107130037261, −7.38471099737186479094268554137, −6.38875723655144207319327471498, −4.74001969428407156303433789350, −3.973605443146439357507384154854, −2.95586306522584012968412434477, −2.02869266521898457107562592482, 1.53901451368238356946784095114, 2.838054639705095419079513946164, 3.96038892799935931461054020452, 4.61797421921458576578176034026, 6.23940529300695843540032210567, 7.15996048816153048325137503343, 8.35462618771022419078386836964, 8.91745337930660369138387853951, 10.695256108870155903821173349070, 11.710336220083705921873333422090, 12.6749182376130627503801398422, 13.39651511644710958176420125890, 14.31979498273766073740207426498, 15.13186908846961162161399865482, 16.12751666000262329506890188350, 16.71985961121462990449759236805, 18.50544889254757958987757232625, 19.5106631413116093671875005566, 20.001339822283670403817850687825, 21.099391045748963599689449978480, 21.57897588658018215551468510559, 22.97048107766703994853222841702, 23.89544275900965876136181734253, 24.38643025620980217057318384117, 25.12183085966138419008283347828

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