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Properties

Label 2-585-65.64-c1-0-7
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

\(L(1)\) \(\approx\) \(0.787655 + 0.787655i\)
\(L(\frac12)\) \(\approx\) \(0.787655 + 0.787655i\)
\(L(\frac{3}{2})\) 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

−10.90840719798179187576143132499, −10.09491540690317150538558088762, −9.205035657676327071547999323178, −7.982407112202709319686642544950, −7.81341848863773237605860557535, −6.26370395043105482593932232570, −5.34921946222101366361750820004, −4.32471463026495286290800673739, −3.30771624253727757590719213019, −1.67829559750508934113047189298, 0.68224825621525657989964919455, 2.30417404089142620941606930762, 4.27441167677038615005817656131, 4.85373335881621089496087951450, 5.35786814989100892331454260842, 7.15971788054442718420816909103, 7.981797626051396352199631059162, 8.770328079607653569085796537423, 9.467286210922240625733817239539, 10.28039557025511575792994619933

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