Properties

Label 2-585-1.1-c1-0-2
Degree $2$
Conductor $585$
Sign $1$
Analytic cond. $4.67124$
Root an. cond. $2.16130$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.77·2-s + 5.71·4-s + 5-s − 2.71·7-s − 10.3·8-s − 2.77·10-s + 2.71·11-s + 13-s + 7.55·14-s + 17.2·16-s + 2.83·17-s − 3.55·19-s + 5.71·20-s − 7.55·22-s + 4.83·23-s + 25-s − 2.77·26-s − 15.5·28-s − 6·29-s + 7.55·31-s − 27.3·32-s − 7.88·34-s − 2.71·35-s − 4.27·37-s + 9.88·38-s − 10.3·40-s − 2.83·41-s + ⋯
L(s)  = 1  − 1.96·2-s + 2.85·4-s + 0.447·5-s − 1.02·7-s − 3.65·8-s − 0.878·10-s + 0.820·11-s + 0.277·13-s + 2.01·14-s + 4.31·16-s + 0.688·17-s − 0.816·19-s + 1.27·20-s − 1.61·22-s + 1.00·23-s + 0.200·25-s − 0.544·26-s − 2.93·28-s − 1.11·29-s + 1.35·31-s − 4.83·32-s − 1.35·34-s − 0.459·35-s − 0.703·37-s + 1.60·38-s − 1.63·40-s − 0.443·41-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(0.6176510005\)
\(L(\frac12)\) \(\approx\) \(0.6176510005\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 - T \)
good2 \( 1 + 2.77T + 2T^{2} \)
7 \( 1 + 2.71T + 7T^{2} \)
11 \( 1 - 2.71T + 11T^{2} \)
17 \( 1 - 2.83T + 17T^{2} \)
19 \( 1 + 3.55T + 19T^{2} \)
23 \( 1 - 4.83T + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 7.55T + 31T^{2} \)
37 \( 1 + 4.27T + 37T^{2} \)
41 \( 1 + 2.83T + 41T^{2} \)
43 \( 1 - 11.1T + 43T^{2} \)
47 \( 1 - 11.5T + 47T^{2} \)
53 \( 1 + 1.16T + 53T^{2} \)
59 \( 1 - 2.11T + 59T^{2} \)
61 \( 1 - 6.60T + 61T^{2} \)
67 \( 1 - 1.88T + 67T^{2} \)
71 \( 1 - 6.71T + 71T^{2} \)
73 \( 1 - 9.11T + 73T^{2} \)
79 \( 1 - 10.2T + 79T^{2} \)
83 \( 1 + 2.11T + 83T^{2} \)
89 \( 1 + 1.16T + 89T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−10.43761722461859984654595425075, −9.618321240138132926479710099222, −9.139975139726218193931962025380, −8.331119901153000427704086331770, −7.20325079308604950179887002220, −6.54581875346392669929961587589, −5.78230755826575569186748267385, −3.51445313017621777717836226457, −2.33808916350099996578520064895, −0.933298169058262645999197157346, 0.933298169058262645999197157346, 2.33808916350099996578520064895, 3.51445313017621777717836226457, 5.78230755826575569186748267385, 6.54581875346392669929961587589, 7.20325079308604950179887002220, 8.331119901153000427704086331770, 9.139975139726218193931962025380, 9.618321240138132926479710099222, 10.43761722461859984654595425075

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