Properties

Label 2-585-1.1-c1-0-4
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 5-s + 3·7-s + 2·10-s + 11-s − 13-s − 6·14-s − 4·16-s + 17-s − 2·19-s − 2·20-s − 2·22-s + 3·23-s + 25-s + 2·26-s + 6·28-s + 2·29-s − 6·31-s + 8·32-s − 2·34-s − 3·35-s + 11·37-s + 4·38-s + 5·41-s + 4·43-s + 2·44-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.447·5-s + 1.13·7-s + 0.632·10-s + 0.301·11-s − 0.277·13-s − 1.60·14-s − 16-s + 0.242·17-s − 0.458·19-s − 0.447·20-s − 0.426·22-s + 0.625·23-s + 1/5·25-s + 0.392·26-s + 1.13·28-s + 0.371·29-s − 1.07·31-s + 1.41·32-s − 0.342·34-s − 0.507·35-s + 1.80·37-s + 0.648·38-s + 0.780·41-s + 0.609·43-s + 0.301·44-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: \(0\)
Selberg data: \((2,\ 585,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7407218622\)
\(L(\frac12)\) \(\approx\) \(0.7407218622\)
\(L(\frac{3}{2})\) 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 \( 1 + p T + p T^{2} \)
7 \( 1 - 3 T + p T^{2} \)
11 \( 1 - T + p T^{2} \)
17 \( 1 - T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 11 T + p T^{2} \)
41 \( 1 - 5 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 11 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 - 17 T + p T^{2} \)
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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

−10.87030732385610107944198759200, −9.599414851193643593710511267052, −8.985183316745324235009539175161, −7.996671602095315844360321530012, −7.64083008607609627592622911960, −6.54697003120115263269660356276, −5.09278902217219401825829455179, −4.09635384552794966459642537511, −2.30897204101866244390058598775, −0.980772475336589212801861022003, 0.980772475336589212801861022003, 2.30897204101866244390058598775, 4.09635384552794966459642537511, 5.09278902217219401825829455179, 6.54697003120115263269660356276, 7.64083008607609627592622911960, 7.996671602095315844360321530012, 8.985183316745324235009539175161, 9.599414851193643593710511267052, 10.87030732385610107944198759200

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