Properties

Label 2-966-1.1-c3-0-3
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 − 10.3·5-s + 6·6-s − 7·7-s − 8·8-s + 9·9-s + 20.6·10-s − 0.432·11-s − 12·12-s + 58.4·13-s + 14·14-s + 31.0·15-s + 16·16-s + 119.·17-s − 18·18-s − 124.·19-s − 41.3·20-s + 21·21-s + 0.865·22-s − 23·23-s + 24·24-s − 18.0·25-s − 116.·26-s − 27·27-s − 28·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.924·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.653·10-s − 0.0118·11-s − 0.288·12-s + 1.24·13-s + 0.267·14-s + 0.533·15-s + 0.250·16-s + 1.70·17-s − 0.235·18-s − 1.50·19-s − 0.462·20-s + 0.218·21-s + 0.00838·22-s − 0.208·23-s + 0.204·24-s − 0.144·25-s − 0.882·26-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

\(L(2)\) \(\approx\) \(0.6448622109\)
\(L(\frac12)\) \(\approx\) \(0.6448622109\)
\(L(\frac{5}{2})\) 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 \( 1 + 10.3T + 125T^{2} \)
11 \( 1 + 0.432T + 1.33e3T^{2} \)
13 \( 1 - 58.4T + 2.19e3T^{2} \)
17 \( 1 - 119.T + 4.91e3T^{2} \)
19 \( 1 + 124.T + 6.85e3T^{2} \)
29 \( 1 + 263.T + 2.43e4T^{2} \)
31 \( 1 + 298.T + 2.97e4T^{2} \)
37 \( 1 - 258.T + 5.06e4T^{2} \)
41 \( 1 - 468.T + 6.89e4T^{2} \)
43 \( 1 + 495.T + 7.95e4T^{2} \)
47 \( 1 + 429.T + 1.03e5T^{2} \)
53 \( 1 - 606.T + 1.48e5T^{2} \)
59 \( 1 - 389.T + 2.05e5T^{2} \)
61 \( 1 + 532.T + 2.26e5T^{2} \)
67 \( 1 - 322.T + 3.00e5T^{2} \)
71 \( 1 + 547.T + 3.57e5T^{2} \)
73 \( 1 + 568.T + 3.89e5T^{2} \)
79 \( 1 - 439.T + 4.93e5T^{2} \)
83 \( 1 - 305.T + 5.71e5T^{2} \)
89 \( 1 + 263.T + 7.04e5T^{2} \)
97 \( 1 + 902.T + 9.12e5T^{2} \)
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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

−9.670341050878305309216171815626, −8.764953112243149852114111221703, −7.913408427728188583737929422153, −7.30960989849689505983168863806, −6.21290761049466337274557026059, −5.59502020340035366561420201095, −4.07098306035199162389493833143, −3.42417832735954392935749311035, −1.75427951228508184461696164894, −0.49601301348566040866557558990, 0.49601301348566040866557558990, 1.75427951228508184461696164894, 3.42417832735954392935749311035, 4.07098306035199162389493833143, 5.59502020340035366561420201095, 6.21290761049466337274557026059, 7.30960989849689505983168863806, 7.913408427728188583737929422153, 8.764953112243149852114111221703, 9.670341050878305309216171815626

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