Properties

Label 2-966-1.1-c1-0-19
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 − 6-s − 7-s − 8-s + 9-s − 2·11-s + 12-s − 6·13-s + 14-s + 16-s + 2·17-s − 18-s − 6·19-s − 21-s + 2·22-s + 23-s − 24-s − 5·25-s + 6·26-s + 27-s − 28-s − 6·29-s − 32-s − 2·33-s − 2·34-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.603·11-s + 0.288·12-s − 1.66·13-s + 0.267·14-s + 1/4·16-s + 0.485·17-s − 0.235·18-s − 1.37·19-s − 0.218·21-s + 0.426·22-s + 0.208·23-s − 0.204·24-s − 25-s + 1.17·26-s + 0.192·27-s − 0.188·28-s − 1.11·29-s − 0.176·32-s − 0.348·33-s − 0.342·34-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 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 2 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−9.592577269260814759793433600070, −8.877959523231252108343638694865, −7.82231004861969744897035522783, −7.43539874264076624837851565651, −6.37358090722116382687867697454, −5.28909785445646003033173706118, −4.09589954641644933076049365904, −2.83475359194195644017632009143, −2.00494291545726260208075635504, 0, 2.00494291545726260208075635504, 2.83475359194195644017632009143, 4.09589954641644933076049365904, 5.28909785445646003033173706118, 6.37358090722116382687867697454, 7.43539874264076624837851565651, 7.82231004861969744897035522783, 8.877959523231252108343638694865, 9.592577269260814759793433600070

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