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Properties

Label 1-287-287.18-r0-0-0
Degree $1$
Conductor $287$
Sign $-0.374 - 0.927i$
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  + (−0.104 − 0.994i)2-s + (−0.5 − 0.866i)3-s + (−0.978 + 0.207i)4-s + (0.669 + 0.743i)5-s + (−0.809 + 0.587i)6-s + (0.309 + 0.951i)8-s + (−0.5 + 0.866i)9-s + (0.669 − 0.743i)10-s + (0.669 − 0.743i)11-s + (0.669 + 0.743i)12-s + (−0.809 + 0.587i)13-s + (0.309 − 0.951i)15-s + (0.913 − 0.406i)16-s + (0.669 − 0.743i)17-s + (0.913 + 0.406i)18-s + (0.913 − 0.406i)19-s + ⋯
L(s)  = 1  + (−0.104 − 0.994i)2-s + (−0.5 − 0.866i)3-s + (−0.978 + 0.207i)4-s + (0.669 + 0.743i)5-s + (−0.809 + 0.587i)6-s + (0.309 + 0.951i)8-s + (−0.5 + 0.866i)9-s + (0.669 − 0.743i)10-s + (0.669 − 0.743i)11-s + (0.669 + 0.743i)12-s + (−0.809 + 0.587i)13-s + (0.309 − 0.951i)15-s + (0.913 − 0.406i)16-s + (0.669 − 0.743i)17-s + (0.913 + 0.406i)18-s + (0.913 − 0.406i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.374 - 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $-0.374 - 0.927i$
Analytic conductor: \(1.33282\)
Root analytic conductor: \(1.33282\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{287} (18, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 287,\ (0:\ ),\ -0.374 - 0.927i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5581667435 - 0.8272453719i\)
\(L(\frac12)\) \(\approx\) \(0.5581667435 - 0.8272453719i\)
\(L(1)\) \(\approx\) \(0.7150544956 - 0.5535135591i\)
\(L(1)\) \(\approx\) \(0.7150544956 - 0.5535135591i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
41 \( 1 \)
good2 \( 1 + (-0.104 - 0.994i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
5 \( 1 + (0.669 + 0.743i)T \)
11 \( 1 + (0.669 - 0.743i)T \)
13 \( 1 + (-0.809 + 0.587i)T \)
17 \( 1 + (0.669 - 0.743i)T \)
19 \( 1 + (0.913 - 0.406i)T \)
23 \( 1 + (-0.104 - 0.994i)T \)
29 \( 1 + (0.309 - 0.951i)T \)
31 \( 1 + (0.669 - 0.743i)T \)
37 \( 1 + (0.669 + 0.743i)T \)
43 \( 1 + (-0.809 + 0.587i)T \)
47 \( 1 + (-0.104 - 0.994i)T \)
53 \( 1 + (-0.978 + 0.207i)T \)
59 \( 1 + (0.913 + 0.406i)T \)
61 \( 1 + (0.913 - 0.406i)T \)
67 \( 1 + (-0.978 + 0.207i)T \)
71 \( 1 + (0.309 + 0.951i)T \)
73 \( 1 + (-0.5 - 0.866i)T \)
79 \( 1 + (-0.5 + 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (0.913 - 0.406i)T \)
97 \( 1 + (0.309 - 0.951i)T \)
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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

−25.56190858924372496506871244296, −25.13860255687335895513671089740, −24.02020692160482576731362053757, −23.17424966355503303252655633207, −22.20957993877133612407615564860, −21.631173097875031210219487809928, −20.480414540952335096925124215403, −19.50810346174185314303677525807, −17.80647448016885042244903310928, −17.510093026379552817314999017781, −16.62100477983922102593568409666, −15.89432318505759066067307392436, −14.82772925779782321083805724753, −14.16568851167092880805968119381, −12.788096775384765434396468119636, −12.01212613293533314794022809440, −10.236989478631194961619270065351, −9.7234985816888428002124396538, −8.875383738741276248773139691753, −7.6409785141893531731448697341, −6.33342200423787885193142058975, −5.39465242284161335835619546165, −4.77657868223200186854683420424, −3.55375601237803628638316225240, −1.24370365304512745907596089711, 0.91484666992465354332572351824, 2.22610218206742601679324804442, 3.0678072500387786557062756555, 4.74444208780875670592904570394, 5.901302589473140197992224963318, 6.94238167020421387178635081836, 8.10515060364937481617399998871, 9.441069843683996947857474925468, 10.243390266655737999274683282383, 11.58582966657547559786982114092, 11.7303191814084287243012352478, 13.14993274443500380534809170494, 13.89336911083154011073178845617, 14.53257301573731998720613153704, 16.57688427880115235896038522293, 17.26987356565850422173905375355, 18.22851482496778424103686761433, 18.83787649130528636321736064307, 19.54147644351924844158587246197, 20.724225665419709297706019199211, 21.896864087107998815861347078163, 22.30448830942134306969266265480, 23.169875881668780158250398290380, 24.34679170835303353729543120024, 25.13931338268685453579025782041

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