Properties

Label 2-585-1.1-c1-0-12
Degree $2$
Conductor $585$
Sign $-1$
Analytic cond. $4.67124$
Root an. cond. $2.16130$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 4-s − 5-s + 2·7-s + 3·8-s + 10-s − 4·11-s − 13-s − 2·14-s − 16-s − 4·17-s + 6·19-s + 20-s + 4·22-s + 25-s + 26-s − 2·28-s − 4·29-s − 10·31-s − 5·32-s + 4·34-s − 2·35-s − 2·37-s − 6·38-s − 3·40-s − 6·41-s − 8·43-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s − 0.447·5-s + 0.755·7-s + 1.06·8-s + 0.316·10-s − 1.20·11-s − 0.277·13-s − 0.534·14-s − 1/4·16-s − 0.970·17-s + 1.37·19-s + 0.223·20-s + 0.852·22-s + 1/5·25-s + 0.196·26-s − 0.377·28-s − 0.742·29-s − 1.79·31-s − 0.883·32-s + 0.685·34-s − 0.338·35-s − 0.328·37-s − 0.973·38-s − 0.474·40-s − 0.937·41-s − 1.21·43-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: yes
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 \( 1 + T + p T^{2} \)
7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 10 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 14 T + p T^{2} \)
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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

−10.19964286858850027386838092871, −9.332189727210250988718420123449, −8.455924495253961688482406234394, −7.77509555057385964870974093324, −7.09926588277152631380243196233, −5.33227627105416183888148278982, −4.80291034122253544985154871503, −3.48108877573719190338767515003, −1.80936597723698845777463745934, 0, 1.80936597723698845777463745934, 3.48108877573719190338767515003, 4.80291034122253544985154871503, 5.33227627105416183888148278982, 7.09926588277152631380243196233, 7.77509555057385964870974093324, 8.455924495253961688482406234394, 9.332189727210250988718420123449, 10.19964286858850027386838092871

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