Properties

Label 2-966-1.1-c3-0-12
Degree $2$
Conductor $966$
Sign $1$
Analytic cond. $56.9958$
Root an. cond. $7.54955$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 4·4-s − 9.55·5-s − 6·6-s + 7·7-s + 8·8-s + 9·9-s − 19.1·10-s − 35.9·11-s − 12·12-s + 6.86·13-s + 14·14-s + 28.6·15-s + 16·16-s − 16.4·17-s + 18·18-s − 27.6·19-s − 38.2·20-s − 21·21-s − 71.8·22-s − 23·23-s − 24·24-s − 33.7·25-s + 13.7·26-s − 27·27-s + 28·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.854·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.604·10-s − 0.985·11-s − 0.288·12-s + 0.146·13-s + 0.267·14-s + 0.493·15-s + 0.250·16-s − 0.235·17-s + 0.235·18-s − 0.333·19-s − 0.427·20-s − 0.218·21-s − 0.696·22-s − 0.208·23-s − 0.204·24-s − 0.270·25-s + 0.103·26-s − 0.192·27-s + 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(56.9958\)
Root analytic conductor: \(7.54955\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(2.004272684\)
\(L(\frac12)\) \(\approx\) \(2.004272684\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
3 \( 1 + 3T \)
7 \( 1 - 7T \)
23 \( 1 + 23T \)
good5 \( 1 + 9.55T + 125T^{2} \)
11 \( 1 + 35.9T + 1.33e3T^{2} \)
13 \( 1 - 6.86T + 2.19e3T^{2} \)
17 \( 1 + 16.4T + 4.91e3T^{2} \)
19 \( 1 + 27.6T + 6.85e3T^{2} \)
29 \( 1 - 307.T + 2.43e4T^{2} \)
31 \( 1 - 278.T + 2.97e4T^{2} \)
37 \( 1 + 225.T + 5.06e4T^{2} \)
41 \( 1 - 151.T + 6.89e4T^{2} \)
43 \( 1 + 434.T + 7.95e4T^{2} \)
47 \( 1 - 281.T + 1.03e5T^{2} \)
53 \( 1 + 337.T + 1.48e5T^{2} \)
59 \( 1 - 744.T + 2.05e5T^{2} \)
61 \( 1 + 345.T + 2.26e5T^{2} \)
67 \( 1 - 465.T + 3.00e5T^{2} \)
71 \( 1 - 956.T + 3.57e5T^{2} \)
73 \( 1 - 536.T + 3.89e5T^{2} \)
79 \( 1 + 541.T + 4.93e5T^{2} \)
83 \( 1 - 352.T + 5.71e5T^{2} \)
89 \( 1 - 1.53e3T + 7.04e5T^{2} \)
97 \( 1 + 226.T + 9.12e5T^{2} \)
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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

−10.00603901130118562836365645886, −8.446488715618765435219059020073, −7.958744088872059774805193424441, −6.93814474744904333447427221608, −6.16414980315537608916367697979, −5.04957028401576268301955165190, −4.52257131753370553412233964009, −3.44590392348046326276590869177, −2.26667085556188909894142384992, −0.69280061746187155936582715679, 0.69280061746187155936582715679, 2.26667085556188909894142384992, 3.44590392348046326276590869177, 4.52257131753370553412233964009, 5.04957028401576268301955165190, 6.16414980315537608916367697979, 6.93814474744904333447427221608, 7.958744088872059774805193424441, 8.446488715618765435219059020073, 10.00603901130118562836365645886

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