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Properties

Label 1-287-287.16-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: $\chi_{287} (16, \cdot )$
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

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

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