Properties

Label 2-966-1.1-c3-0-51
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 3·3-s + 4·4-s − 2.20·5-s − 6·6-s − 7·7-s − 8·8-s + 9·9-s + 4.40·10-s + 29.6·11-s + 12·12-s + 23.3·13-s + 14·14-s − 6.61·15-s + 16·16-s − 96.0·17-s − 18·18-s − 56.0·19-s − 8.81·20-s − 21·21-s − 59.3·22-s − 23·23-s − 24·24-s − 120.·25-s − 46.7·26-s + 27·27-s − 28·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.197·5-s − 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.139·10-s + 0.813·11-s + 0.288·12-s + 0.499·13-s + 0.267·14-s − 0.113·15-s + 0.250·16-s − 1.37·17-s − 0.235·18-s − 0.676·19-s − 0.0985·20-s − 0.218·21-s − 0.575·22-s − 0.208·23-s − 0.204·24-s − 0.961·25-s − 0.352·26-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: \(1\)
Selberg data: \((2,\ 966,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 2.20T + 125T^{2} \)
11 \( 1 - 29.6T + 1.33e3T^{2} \)
13 \( 1 - 23.3T + 2.19e3T^{2} \)
17 \( 1 + 96.0T + 4.91e3T^{2} \)
19 \( 1 + 56.0T + 6.85e3T^{2} \)
29 \( 1 - 210.T + 2.43e4T^{2} \)
31 \( 1 - 129.T + 2.97e4T^{2} \)
37 \( 1 + 108.T + 5.06e4T^{2} \)
41 \( 1 + 195.T + 6.89e4T^{2} \)
43 \( 1 + 224.T + 7.95e4T^{2} \)
47 \( 1 - 126.T + 1.03e5T^{2} \)
53 \( 1 - 382.T + 1.48e5T^{2} \)
59 \( 1 + 648.T + 2.05e5T^{2} \)
61 \( 1 + 47.6T + 2.26e5T^{2} \)
67 \( 1 + 520.T + 3.00e5T^{2} \)
71 \( 1 + 994.T + 3.57e5T^{2} \)
73 \( 1 + 76.6T + 3.89e5T^{2} \)
79 \( 1 + 692.T + 4.93e5T^{2} \)
83 \( 1 - 1.28e3T + 5.71e5T^{2} \)
89 \( 1 - 564.T + 7.04e5T^{2} \)
97 \( 1 + 428.T + 9.12e5T^{2} \)
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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

−8.930272509626874648180241629934, −8.680356946734554945619299539364, −7.69571068465633472118578979824, −6.67756507791107982977463596390, −6.20583204534692070214944982954, −4.57334580757735090486056625340, −3.68275665359415063708540783624, −2.54646127836588584059394727671, −1.45178833613447801449564461512, 0, 1.45178833613447801449564461512, 2.54646127836588584059394727671, 3.68275665359415063708540783624, 4.57334580757735090486056625340, 6.20583204534692070214944982954, 6.67756507791107982977463596390, 7.69571068465633472118578979824, 8.680356946734554945619299539364, 8.930272509626874648180241629934

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