Properties

Degree $2$
Conductor $966$
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 + 8-s + 9-s − 2·10-s − 4·11-s − 12-s − 2·13-s + 14-s + 2·15-s + 16-s − 6·17-s + 18-s + 4·19-s − 2·20-s − 21-s − 4·22-s − 23-s − 24-s − 25-s − 2·26-s − 27-s + 28-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 + 0.353·8-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 − 1.45·17-s + 0.235·18-s + 0.917·19-s − 0.447·20-s − 0.218·21-s − 0.852·22-s − 0.208·23-s − 0.204·24-s − 1/5·25-s − 0.392·26-s − 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{966} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
3 \( 1 + T \)
7 \( 1 - T \)
23 \( 1 + T \)
good5 \( 1 + 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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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

−19.40035304637725, −18.20418028225171, −18.11923015265734, −16.92606622041165, −16.26205571290245, −15.55196734371521, −15.17762909059791, −14.26816094016128, −13.22275910050595, −12.88739362436172, −11.76777285228057, −11.46967342888755, −10.70069666134580, −9.836103353240370, −8.598497618710542, −7.606119472915856, −7.177309883486510, −5.966664745632519, −5.071317475686980, −4.447154374285747, −3.339454574646301, −2.058633334656759, 0, 2.058633334656759, 3.339454574646301, 4.447154374285747, 5.071317475686980, 5.966664745632519, 7.177309883486510, 7.606119472915856, 8.598497618710542, 9.836103353240370, 10.70069666134580, 11.46967342888755, 11.76777285228057, 12.88739362436172, 13.22275910050595, 14.26816094016128, 15.17762909059791, 15.55196734371521, 16.26205571290245, 16.92606622041165, 18.11923015265734, 18.20418028225171, 19.40035304637725

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