Properties

Label 2-966-1.1-c1-0-9
Degree $2$
Conductor $966$
Sign $1$
Analytic cond. $7.71354$
Root an. cond. $2.77732$
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-s − 3-s + 4-s + 5-s − 6-s + 7-s + 8-s + 9-s + 10-s − 12-s − 13-s + 14-s − 15-s + 16-s + 4·17-s + 18-s + 4·19-s + 20-s − 21-s + 23-s − 24-s − 4·25-s − 26-s − 27-s + 28-s − 29-s − 30-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.258·15-s + 1/4·16-s + 0.970·17-s + 0.235·18-s + 0.917·19-s + 0.223·20-s − 0.218·21-s + 0.208·23-s − 0.204·24-s − 4/5·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s − 0.185·29-s − 0.182·30-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: $1$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{966} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.396153591\)
\(L(\frac12)\) \(\approx\) \(2.396153591\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 + T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + T + p T^{2} \)
31 \( 1 - 10 T + p T^{2} \)
37 \( 1 + 5 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 4 T + p T^{2} \)
79 \( 1 - 12 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 + T + p T^{2} \)
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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

−10.12312561609253227089628948996, −9.447878334573988174395171457981, −8.122604412468263161300911404608, −7.36326170221295729421698558385, −6.38853493402900646955242584366, −5.56292383619534735128047063888, −4.94611903624845462114112658465, −3.84915047200430085189740243685, −2.63399820894633158776953237185, −1.26483261476793153231223788462, 1.26483261476793153231223788462, 2.63399820894633158776953237185, 3.84915047200430085189740243685, 4.94611903624845462114112658465, 5.56292383619534735128047063888, 6.38853493402900646955242584366, 7.36326170221295729421698558385, 8.122604412468263161300911404608, 9.447878334573988174395171457981, 10.12312561609253227089628948996

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