Properties

Degree $2$
Conductor $966$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

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 − 2·20-s − 21-s + 4·22-s + 23-s + 24-s − 25-s + 2·26-s + 27-s − 28-s − 2·29-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 + 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 − 0.447·20-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 − 0.371·29-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: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.785859174\)
\(L(\frac12)\) \(\approx\) \(2.785859174\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
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 \( 1 + 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−19.59898598328709, −19.15243637764348, −18.45713555924502, −17.28482013004122, −16.48532021785977, −15.99411594496336, −15.12325736702405, −14.64921142003217, −13.92136289670841, −13.19544569932966, −12.27803632902950, −11.83355038125173, −11.02155838848655, −9.968594132488272, −9.190747247293543, −8.181256656550651, −7.520501337946085, −6.594287683128709, −5.733340524802486, −4.393767706045575, −3.721751174960615, −2.977987879245730, −1.317545632041997, 1.317545632041997, 2.977987879245730, 3.721751174960615, 4.393767706045575, 5.733340524802486, 6.594287683128709, 7.520501337946085, 8.181256656550651, 9.190747247293543, 9.968594132488272, 11.02155838848655, 11.83355038125173, 12.27803632902950, 13.19544569932966, 13.92136289670841, 14.64921142003217, 15.12325736702405, 15.99411594496336, 16.48532021785977, 17.28482013004122, 18.45713555924502, 19.15243637764348, 19.59898598328709

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