Properties

Label 2-585-1.1-c1-0-9
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  + 1.73·2-s + 0.999·4-s + 5-s + 2·7-s − 1.73·8-s + 1.73·10-s + 4.73·11-s + 13-s + 3.46·14-s − 5·16-s + 3.46·17-s − 6.19·19-s + 0.999·20-s + 8.19·22-s − 1.26·23-s + 25-s + 1.73·26-s + 1.99·28-s + 2.53·29-s + 10.1·31-s − 5.19·32-s + 5.99·34-s + 2·35-s − 4·37-s − 10.7·38-s − 1.73·40-s − 3.46·41-s + ⋯
L(s)  = 1  + 1.22·2-s + 0.499·4-s + 0.447·5-s + 0.755·7-s − 0.612·8-s + 0.547·10-s + 1.42·11-s + 0.277·13-s + 0.925·14-s − 1.25·16-s + 0.840·17-s − 1.42·19-s + 0.223·20-s + 1.74·22-s − 0.264·23-s + 0.200·25-s + 0.339·26-s + 0.377·28-s + 0.470·29-s + 1.83·31-s − 0.918·32-s + 1.02·34-s + 0.338·35-s − 0.657·37-s − 1.74·38-s − 0.273·40-s − 0.541·41-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: $\chi_{585} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.984453741\)
\(L(\frac12)\) \(\approx\) \(2.984453741\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.73T + 2T^{2} \)
7 \( 1 - 2T + 7T^{2} \)
11 \( 1 - 4.73T + 11T^{2} \)
17 \( 1 - 3.46T + 17T^{2} \)
19 \( 1 + 6.19T + 19T^{2} \)
23 \( 1 + 1.26T + 23T^{2} \)
29 \( 1 - 2.53T + 29T^{2} \)
31 \( 1 - 10.1T + 31T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + 3.46T + 41T^{2} \)
43 \( 1 + 0.196T + 43T^{2} \)
47 \( 1 + 6T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + 9.12T + 59T^{2} \)
61 \( 1 + 8.39T + 61T^{2} \)
67 \( 1 - 6.39T + 67T^{2} \)
71 \( 1 + 4.73T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 + 8.39T + 79T^{2} \)
83 \( 1 - 6T + 83T^{2} \)
89 \( 1 - 12.9T + 89T^{2} \)
97 \( 1 - 2T + 97T^{2} \)
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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

−10.94472824657291446880371885590, −9.877348407940449699969248205996, −8.904475350883832398689412944219, −8.113862234208344136020896150558, −6.56456973101159506347002258038, −6.15119478756246662288907107930, −4.92456319302782996777442433173, −4.25102037691181755072506709673, −3.13978927529927978171053329719, −1.63717590887318719464495311003, 1.63717590887318719464495311003, 3.13978927529927978171053329719, 4.25102037691181755072506709673, 4.92456319302782996777442433173, 6.15119478756246662288907107930, 6.56456973101159506347002258038, 8.113862234208344136020896150558, 8.904475350883832398689412944219, 9.877348407940449699969248205996, 10.94472824657291446880371885590

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