Properties

Label 2-585-1.1-c1-0-19
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.56·2-s + 0.438·4-s − 5-s − 4.56·7-s − 2.43·8-s − 1.56·10-s − 2.56·11-s − 13-s − 7.12·14-s − 4.68·16-s + 2.56·17-s + 3.12·19-s − 0.438·20-s − 4·22-s − 6.56·23-s + 25-s − 1.56·26-s − 1.99·28-s − 1.12·29-s + 6·31-s − 2.43·32-s + 4·34-s + 4.56·35-s + 1.68·37-s + 4.87·38-s + 2.43·40-s + 0.561·41-s + ⋯
L(s)  = 1  + 1.10·2-s + 0.219·4-s − 0.447·5-s − 1.72·7-s − 0.862·8-s − 0.493·10-s − 0.772·11-s − 0.277·13-s − 1.90·14-s − 1.17·16-s + 0.621·17-s + 0.716·19-s − 0.0980·20-s − 0.852·22-s − 1.36·23-s + 0.200·25-s − 0.306·26-s − 0.377·28-s − 0.208·29-s + 1.07·31-s − 0.431·32-s + 0.685·34-s + 0.771·35-s + 0.276·37-s + 0.791·38-s + 0.385·40-s + 0.0876·41-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: \(1\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.56T + 2T^{2} \)
7 \( 1 + 4.56T + 7T^{2} \)
11 \( 1 + 2.56T + 11T^{2} \)
17 \( 1 - 2.56T + 17T^{2} \)
19 \( 1 - 3.12T + 19T^{2} \)
23 \( 1 + 6.56T + 23T^{2} \)
29 \( 1 + 1.12T + 29T^{2} \)
31 \( 1 - 6T + 31T^{2} \)
37 \( 1 - 1.68T + 37T^{2} \)
41 \( 1 - 0.561T + 41T^{2} \)
43 \( 1 + 5.12T + 43T^{2} \)
47 \( 1 + 2.87T + 47T^{2} \)
53 \( 1 + 7.68T + 53T^{2} \)
59 \( 1 + 12T + 59T^{2} \)
61 \( 1 - 5.68T + 61T^{2} \)
67 \( 1 - 13.3T + 67T^{2} \)
71 \( 1 + 14.5T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 + 7.68T + 79T^{2} \)
83 \( 1 + 16.4T + 83T^{2} \)
89 \( 1 - 1.68T + 89T^{2} \)
97 \( 1 + 11.9T + 97T^{2} \)
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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

−10.04323222908775155559777351237, −9.693664612531849309713983016073, −8.460648932664054176160298451417, −7.38527339122787260506101003182, −6.33191804757917848346117094712, −5.64656296920201328406111053153, −4.51937609989202770452600308295, −3.46360417048540056927518683779, −2.82342396833124201444253741024, 0, 2.82342396833124201444253741024, 3.46360417048540056927518683779, 4.51937609989202770452600308295, 5.64656296920201328406111053153, 6.33191804757917848346117094712, 7.38527339122787260506101003182, 8.460648932664054176160298451417, 9.693664612531849309713983016073, 10.04323222908775155559777351237

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