Properties

Label 2-966-1.1-c3-0-2
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 − 18.2·5-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s − 36.5·10-s − 49.0·11-s − 12·12-s − 87.3·13-s − 14·14-s + 54.8·15-s + 16·16-s + 60.5·17-s + 18·18-s − 124.·19-s − 73.1·20-s + 21·21-s − 98.0·22-s + 23·23-s − 24·24-s + 209.·25-s − 174.·26-s − 27·27-s − 28·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.63·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 1.15·10-s − 1.34·11-s − 0.288·12-s − 1.86·13-s − 0.267·14-s + 0.943·15-s + 0.250·16-s + 0.864·17-s + 0.235·18-s − 1.50·19-s − 0.817·20-s + 0.218·21-s − 0.950·22-s + 0.208·23-s − 0.204·24-s + 1.67·25-s − 1.31·26-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

\(L(2)\) \(\approx\) \(0.6512763820\)
\(L(\frac12)\) \(\approx\) \(0.6512763820\)
\(L(\frac{5}{2})\) 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 \( 1 + 18.2T + 125T^{2} \)
11 \( 1 + 49.0T + 1.33e3T^{2} \)
13 \( 1 + 87.3T + 2.19e3T^{2} \)
17 \( 1 - 60.5T + 4.91e3T^{2} \)
19 \( 1 + 124.T + 6.85e3T^{2} \)
29 \( 1 + 87.7T + 2.43e4T^{2} \)
31 \( 1 - 296.T + 2.97e4T^{2} \)
37 \( 1 + 174.T + 5.06e4T^{2} \)
41 \( 1 + 47.2T + 6.89e4T^{2} \)
43 \( 1 - 181.T + 7.95e4T^{2} \)
47 \( 1 + 390.T + 1.03e5T^{2} \)
53 \( 1 - 489.T + 1.48e5T^{2} \)
59 \( 1 + 152.T + 2.05e5T^{2} \)
61 \( 1 + 185.T + 2.26e5T^{2} \)
67 \( 1 + 10.5T + 3.00e5T^{2} \)
71 \( 1 + 680.T + 3.57e5T^{2} \)
73 \( 1 + 1.09e3T + 3.89e5T^{2} \)
79 \( 1 - 898.T + 4.93e5T^{2} \)
83 \( 1 - 283.T + 5.71e5T^{2} \)
89 \( 1 - 618.T + 7.04e5T^{2} \)
97 \( 1 - 336.T + 9.12e5T^{2} \)
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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

−10.06741168276335474637515206465, −8.538070019252671101973667857838, −7.60820623517762665688754626894, −7.26172045685715203619171578517, −6.13470701473656430136511088254, −4.96304836166834251747489663419, −4.54368326268435964805623320797, −3.39260067678410510459112624678, −2.44081615697129816847909284853, −0.37428751510762374687025625249, 0.37428751510762374687025625249, 2.44081615697129816847909284853, 3.39260067678410510459112624678, 4.54368326268435964805623320797, 4.96304836166834251747489663419, 6.13470701473656430136511088254, 7.26172045685715203619171578517, 7.60820623517762665688754626894, 8.538070019252671101973667857838, 10.06741168276335474637515206465

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