Properties

Degree 1
Conductor $ 7 \cdot 41 $
Sign $-0.374 + 0.927i$
Motivic weight 0
Primitive yes
Self-dual no
Analytic rank 0

Related objects

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Normalization:  

Dirichlet series

L(χ,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,χ)  = 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

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

Invariants

\( d \)  =  \(1\)
\( N \)  =  \(287\)    =    \(7 \cdot 41\)
\( \varepsilon \)  =  $-0.374 + 0.927i$
motivic weight  =  \(0\)
character  :  $\chi_{287} (16, \cdot )$
Sato-Tate  :  $\mu(15)$
primitive  :  yes
self-dual  :  no
analytic rank  =  0
Selberg data  =  $(1,\ 287,\ (0:\ ),\ -0.374 + 0.927i)$
$L(\chi,\frac{1}{2})$  $\approx$  $0.5581667435 + 0.8272453719i$
$L(\frac12,\chi)$  $\approx$  $0.5581667435 + 0.8272453719i$
$L(\chi,1)$  $\approx$  0.7150544956 + 0.5535135591i
$L(1,\chi)$  $\approx$  0.7150544956 + 0.5535135591i

Euler product

\[\begin{aligned}L(\chi,s) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]
\[\begin{aligned}L(s,\chi) = \prod_p (1- \chi(p) p^{-s})^{-1}\end{aligned}\]

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