Properties

Label 2-966-1.1-c1-0-17
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 + 3·5-s + 6-s − 7-s + 8-s + 9-s + 3·10-s + 4·11-s + 12-s − 3·13-s − 14-s + 3·15-s + 16-s − 4·17-s + 18-s + 3·20-s − 21-s + 4·22-s + 23-s + 24-s + 4·25-s − 3·26-s + 27-s − 28-s + 3·29-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 1.34·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.948·10-s + 1.20·11-s + 0.288·12-s − 0.832·13-s − 0.267·14-s + 0.774·15-s + 1/4·16-s − 0.970·17-s + 0.235·18-s + 0.670·20-s − 0.218·21-s + 0.852·22-s + 0.208·23-s + 0.204·24-s + 4/5·25-s − 0.588·26-s + 0.192·27-s − 0.188·28-s + 0.557·29-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: $1$
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

\(L(1)\) \(\approx\) \(3.593527008\)
\(L(\frac12)\) \(\approx\) \(3.593527008\)
\(L(\frac{3}{2})\) 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 \( 1 - 3 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 + 9 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 + 3 T + p T^{2} \)
47 \( 1 + 7 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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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

−9.843827071515005252615180187856, −9.345059082219016767573214421285, −8.533329024037475026270064025246, −7.11624966976560567011680890410, −6.61433587019532483612650095921, −5.70303300160941996822497347087, −4.72846979566270558944303333386, −3.67780852772662808140519843755, −2.54506917140467324126940654920, −1.68100316040744687359155389371, 1.68100316040744687359155389371, 2.54506917140467324126940654920, 3.67780852772662808140519843755, 4.72846979566270558944303333386, 5.70303300160941996822497347087, 6.61433587019532483612650095921, 7.11624966976560567011680890410, 8.533329024037475026270064025246, 9.345059082219016767573214421285, 9.843827071515005252615180187856

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