Properties

Label 2-966-1.1-c1-0-13
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 4·5-s + 6-s + 7-s − 8-s + 9-s + 4·10-s + 2·11-s − 12-s + 2·13-s − 14-s + 4·15-s + 16-s + 2·17-s − 18-s − 2·19-s − 4·20-s − 21-s − 2·22-s + 23-s + 24-s + 11·25-s − 2·26-s − 27-s + 28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s − 1.78·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.26·10-s + 0.603·11-s − 0.288·12-s + 0.554·13-s − 0.267·14-s + 1.03·15-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 0.458·19-s − 0.894·20-s − 0.218·21-s − 0.426·22-s + 0.208·23-s + 0.204·24-s + 11/5·25-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: $-1$
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: \(1\)
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 \( 1 + 4 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
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

−9.590145272834824356845533988063, −8.517749678089344145487901033794, −8.057806690325987987133750230050, −7.17223625973566331940704514276, −6.47795508864117195965745614559, −5.18705288465943249189670817747, −4.12890850837478174257592345990, −3.32637588427684486740674767745, −1.41523106223574871080970468543, 0, 1.41523106223574871080970468543, 3.32637588427684486740674767745, 4.12890850837478174257592345990, 5.18705288465943249189670817747, 6.47795508864117195965745614559, 7.17223625973566331940704514276, 8.057806690325987987133750230050, 8.517749678089344145487901033794, 9.590145272834824356845533988063

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