Properties

Label 2-966-1.1-c3-0-47
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 − 13.6·5-s − 6·6-s − 7·7-s + 8·8-s + 9·9-s − 27.3·10-s − 2.24·11-s − 12·12-s + 45.7·13-s − 14·14-s + 41.0·15-s + 16·16-s + 4.13·17-s + 18·18-s + 133.·19-s − 54.6·20-s + 21·21-s − 4.49·22-s − 23·23-s − 24·24-s + 61.8·25-s + 91.4·26-s − 27·27-s − 28·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.22·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s − 0.864·10-s − 0.0616·11-s − 0.288·12-s + 0.975·13-s − 0.267·14-s + 0.705·15-s + 0.250·16-s + 0.0589·17-s + 0.235·18-s + 1.60·19-s − 0.611·20-s + 0.218·21-s − 0.0435·22-s − 0.208·23-s − 0.204·24-s + 0.495·25-s + 0.690·26-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 \( 1 + 13.6T + 125T^{2} \)
11 \( 1 + 2.24T + 1.33e3T^{2} \)
13 \( 1 - 45.7T + 2.19e3T^{2} \)
17 \( 1 - 4.13T + 4.91e3T^{2} \)
19 \( 1 - 133.T + 6.85e3T^{2} \)
29 \( 1 + 197.T + 2.43e4T^{2} \)
31 \( 1 - 116.T + 2.97e4T^{2} \)
37 \( 1 - 54.4T + 5.06e4T^{2} \)
41 \( 1 + 234.T + 6.89e4T^{2} \)
43 \( 1 + 266.T + 7.95e4T^{2} \)
47 \( 1 + 341.T + 1.03e5T^{2} \)
53 \( 1 - 114.T + 1.48e5T^{2} \)
59 \( 1 + 292.T + 2.05e5T^{2} \)
61 \( 1 + 648.T + 2.26e5T^{2} \)
67 \( 1 + 329.T + 3.00e5T^{2} \)
71 \( 1 + 57.5T + 3.57e5T^{2} \)
73 \( 1 + 536.T + 3.89e5T^{2} \)
79 \( 1 + 245.T + 4.93e5T^{2} \)
83 \( 1 - 481.T + 5.71e5T^{2} \)
89 \( 1 - 750.T + 7.04e5T^{2} \)
97 \( 1 - 1.25e3T + 9.12e5T^{2} \)
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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

−9.280199768865867791787383303529, −8.094629512052739604491507519291, −7.45335831784123515259125260654, −6.55625697291784403896443533897, −5.68845170836781942887092469321, −4.77072708144203366688489064690, −3.77500726212173992840142800230, −3.16786511722620630307983466101, −1.36882015803993116230536992127, 0, 1.36882015803993116230536992127, 3.16786511722620630307983466101, 3.77500726212173992840142800230, 4.77072708144203366688489064690, 5.68845170836781942887092469321, 6.55625697291784403896443533897, 7.45335831784123515259125260654, 8.094629512052739604491507519291, 9.280199768865867791787383303529

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