Properties

Degree $2$
Conductor $609$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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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: $-1$
Motivic weight: \(1\)
Character: $\chi_{609} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
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 \( 1 + 6 T + p T^{2} \)
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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

−19.88556999175148, −19.32167380805058, −18.92463800586466, −17.88068363528671, −17.40785963064592, −16.71119394741763, −16.13656810045602, −14.97943607291374, −14.45554775752049, −13.45766310949034, −12.38781788539849, −11.86992263739325, −10.95889957996791, −10.23880777751835, −9.239297261474231, −8.550313716674631, −7.609109951986150, −6.905815644073383, −5.548440727819883, −4.469491937092364, −3.763851777572223, −1.560619303765826, 0, 1.560619303765826, 3.763851777572223, 4.469491937092364, 5.548440727819883, 6.905815644073383, 7.609109951986150, 8.550313716674631, 9.239297261474231, 10.23880777751835, 10.95889957996791, 11.86992263739325, 12.38781788539849, 13.45766310949034, 14.45554775752049, 14.97943607291374, 16.13656810045602, 16.71119394741763, 17.40785963064592, 17.88068363528671, 18.92463800586466, 19.32167380805058, 19.88556999175148

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