Properties

Label 2-966-1.1-c3-0-4
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 + 10.1·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s − 20.2·10-s − 47.7·11-s − 12·12-s − 77.2·13-s + 14·14-s − 30.4·15-s + 16·16-s − 26.2·17-s − 18·18-s + 54.9·19-s + 40.5·20-s + 21·21-s + 95.5·22-s − 23·23-s + 24·24-s − 22.0·25-s + 154.·26-s − 27·27-s − 28·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.907·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.641·10-s − 1.30·11-s − 0.288·12-s − 1.64·13-s + 0.267·14-s − 0.523·15-s + 0.250·16-s − 0.374·17-s − 0.235·18-s + 0.663·19-s + 0.453·20-s + 0.218·21-s + 0.925·22-s − 0.208·23-s + 0.204·24-s − 0.176·25-s + 1.16·26-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

\(L(2)\) \(\approx\) \(0.7885929251\)
\(L(\frac12)\) \(\approx\) \(0.7885929251\)
\(L(\frac{5}{2})\) 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 \( 1 - 10.1T + 125T^{2} \)
11 \( 1 + 47.7T + 1.33e3T^{2} \)
13 \( 1 + 77.2T + 2.19e3T^{2} \)
17 \( 1 + 26.2T + 4.91e3T^{2} \)
19 \( 1 - 54.9T + 6.85e3T^{2} \)
29 \( 1 + 52.9T + 2.43e4T^{2} \)
31 \( 1 + 97.0T + 2.97e4T^{2} \)
37 \( 1 - 295.T + 5.06e4T^{2} \)
41 \( 1 - 365.T + 6.89e4T^{2} \)
43 \( 1 - 392.T + 7.95e4T^{2} \)
47 \( 1 - 79.9T + 1.03e5T^{2} \)
53 \( 1 + 146.T + 1.48e5T^{2} \)
59 \( 1 + 19.0T + 2.05e5T^{2} \)
61 \( 1 - 89.2T + 2.26e5T^{2} \)
67 \( 1 + 1.02e3T + 3.00e5T^{2} \)
71 \( 1 + 121.T + 3.57e5T^{2} \)
73 \( 1 + 749.T + 3.89e5T^{2} \)
79 \( 1 - 548.T + 4.93e5T^{2} \)
83 \( 1 - 73.0T + 5.71e5T^{2} \)
89 \( 1 - 1.08e3T + 7.04e5T^{2} \)
97 \( 1 - 724.T + 9.12e5T^{2} \)
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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

−9.675191772094885954077502829777, −9.151464891674154761364474722143, −7.67099565151689244810104949897, −7.37423084546189453805903575485, −6.10190723556114739509246801492, −5.54837728029661330739861253223, −4.55234591994062507145449997259, −2.80943022359566815360018308203, −2.07540100124240468021451588113, −0.51819041370872802208889147314, 0.51819041370872802208889147314, 2.07540100124240468021451588113, 2.80943022359566815360018308203, 4.55234591994062507145449997259, 5.54837728029661330739861253223, 6.10190723556114739509246801492, 7.37423084546189453805903575485, 7.67099565151689244810104949897, 9.151464891674154761364474722143, 9.675191772094885954077502829777

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