Properties

Label 2-609-1.1-c1-0-12
Degree $2$
Conductor $609$
Sign $-1$
Analytic cond. $4.86288$
Root an. cond. $2.20519$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s − 4-s − 2·5-s + 6-s + 7-s + 3·8-s + 9-s + 2·10-s + 4·11-s + 12-s − 2·13-s − 14-s + 2·15-s − 16-s + 2·17-s − 18-s − 4·19-s + 2·20-s − 21-s − 4·22-s − 3·24-s − 25-s + 2·26-s − 27-s − 28-s + 29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.377·7-s + 1.06·8-s + 1/3·9-s + 0.632·10-s + 1.20·11-s + 0.288·12-s − 0.554·13-s − 0.267·14-s + 0.516·15-s − 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.917·19-s + 0.447·20-s − 0.218·21-s − 0.852·22-s − 0.612·24-s − 1/5·25-s + 0.392·26-s − 0.192·27-s − 0.188·28-s + 0.185·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 609 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 609 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(609\)    =    \(3 \cdot 7 \cdot 29\)
Sign: $-1$
Analytic conductor: \(4.86288\)
Root analytic conductor: \(2.20519\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 609,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
7 \( 1 - T \)
29 \( 1 - T \)
good2 \( 1 + T + p T^{2} \)
5 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.23880777751834639911403853835, −9.239297261474231391016729539472, −8.550313716674631187231272371318, −7.60910995198615038777281064257, −6.90581564407338290859312696888, −5.54844072781988270714879864647, −4.46949193709236381800050091542, −3.76385177757222278085780912103, −1.56061930376582561328528487609, 0, 1.56061930376582561328528487609, 3.76385177757222278085780912103, 4.46949193709236381800050091542, 5.54844072781988270714879864647, 6.90581564407338290859312696888, 7.60910995198615038777281064257, 8.550313716674631187231272371318, 9.239297261474231391016729539472, 10.23880777751834639911403853835

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