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Properties

Label 2-585-65.64-c1-0-2
Degree $2$
Conductor $585$
Sign $-i$
Analytic cond. $4.67124$
Root an. cond. $2.16130$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·4-s − 2.23i·5-s − 3.60·7-s + 2.23i·11-s + 3.60·13-s + 4·16-s + 8.06i·17-s + 4.47i·20-s + 8.06i·23-s − 5.00·25-s + 7.21·28-s + 8.06i·35-s − 3.60·37-s − 11.1i·41-s − 4.47i·44-s + ⋯
L(s)  = 1  − 4-s − 0.999i·5-s − 1.36·7-s + 0.674i·11-s + 1.00·13-s + 16-s + 1.95i·17-s + 0.999i·20-s + 1.68i·23-s − 1.00·25-s + 1.36·28-s + 1.36i·35-s − 0.592·37-s − 1.74i·41-s − 0.674i·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(585\)    =    \(3^{2} \cdot 5 \cdot 13\)
Sign: $-i$
Analytic conductor: \(4.67124\)
Root analytic conductor: \(2.16130\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{585} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.421540 + 0.421540i\)
\(L(\frac12)\) \(\approx\) \(0.421540 + 0.421540i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + 2.23iT \)
13 \( 1 - 3.60T \)
good2 \( 1 + 2T^{2} \)
7 \( 1 + 3.60T + 7T^{2} \)
11 \( 1 - 2.23iT - 11T^{2} \)
17 \( 1 - 8.06iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 8.06iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 3.60T + 37T^{2} \)
41 \( 1 + 11.1iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 8.06iT - 53T^{2} \)
59 \( 1 - 8.94iT - 59T^{2} \)
61 \( 1 + 7T + 61T^{2} \)
67 \( 1 + 14.4T + 67T^{2} \)
71 \( 1 - 15.6iT - 71T^{2} \)
73 \( 1 - 7.21T + 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 2.23iT - 89T^{2} \)
97 \( 1 + 3.60T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.64959093880742045827143833002, −9.874525492569226838286547095567, −9.117041957776312102367674665629, −8.567924488198617211728970963846, −7.51166875015286550108672545056, −6.11926811965491154687413270640, −5.48767134600582876150675813109, −4.12964500371094541039026739454, −3.58807539949549318511065618997, −1.43178131077830570229078081033, 0.36881477182371913175736337206, 2.91522951295656554951821953702, 3.52111941443955530834076750881, 4.82696543800890614048691086724, 6.12852378394196762989542180482, 6.66370143666822466027739805187, 7.87519005697604201429292484777, 8.913686922027174635693349457405, 9.598693601273356216215560229931, 10.35953442359925804597739457524

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