Properties

Label 2-966-1.1-c1-0-8
Degree $2$
Conductor $966$
Sign $1$
Analytic cond. $7.71354$
Root an. cond. $2.77732$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s + 4-s + 1.37·5-s − 6-s − 7-s + 8-s + 9-s + 1.37·10-s + 4·11-s − 12-s + 1.37·13-s − 14-s − 1.37·15-s + 16-s − 4.74·17-s + 18-s + 4·19-s + 1.37·20-s + 21-s + 4·22-s − 23-s − 24-s − 3.11·25-s + 1.37·26-s − 27-s − 28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.613·5-s − 0.408·6-s − 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.433·10-s + 1.20·11-s − 0.288·12-s + 0.380·13-s − 0.267·14-s − 0.354·15-s + 0.250·16-s − 1.15·17-s + 0.235·18-s + 0.917·19-s + 0.306·20-s + 0.218·21-s + 0.852·22-s − 0.208·23-s − 0.204·24-s − 0.623·25-s + 0.269·26-s − 0.192·27-s − 0.188·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(966\)    =    \(2 \cdot 3 \cdot 7 \cdot 23\)
Sign: $1$
Analytic conductor: \(7.71354\)
Root analytic conductor: \(2.77732\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 966,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.381763565\)
\(L(\frac12)\) \(\approx\) \(2.381763565\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 + T \)
7 \( 1 + T \)
23 \( 1 + T \)
good5 \( 1 - 1.37T + 5T^{2} \)
11 \( 1 - 4T + 11T^{2} \)
13 \( 1 - 1.37T + 13T^{2} \)
17 \( 1 + 4.74T + 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
29 \( 1 - 9.37T + 29T^{2} \)
31 \( 1 - 6.74T + 31T^{2} \)
37 \( 1 - 2.62T + 37T^{2} \)
41 \( 1 + 8.11T + 41T^{2} \)
43 \( 1 - 6.11T + 43T^{2} \)
47 \( 1 - 4.62T + 47T^{2} \)
53 \( 1 - 4.74T + 53T^{2} \)
59 \( 1 + 2.74T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 14.7T + 71T^{2} \)
73 \( 1 + 12.7T + 73T^{2} \)
79 \( 1 + 13.4T + 79T^{2} \)
83 \( 1 - 4T + 83T^{2} \)
89 \( 1 - 7.48T + 89T^{2} \)
97 \( 1 + 9.37T + 97T^{2} \)
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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

−10.09124647598437864200465143322, −9.366345500486942964371074804205, −8.383255742330188319790166595633, −7.05251170518586249791462921764, −6.41104774704304517006593120815, −5.84065065123578747693183381305, −4.72069752741763663242035470103, −3.91111775644876576270518825142, −2.63902010255096877862274483980, −1.25781928333410151899553950102, 1.25781928333410151899553950102, 2.63902010255096877862274483980, 3.91111775644876576270518825142, 4.72069752741763663242035470103, 5.84065065123578747693183381305, 6.41104774704304517006593120815, 7.05251170518586249791462921764, 8.383255742330188319790166595633, 9.366345500486942964371074804205, 10.09124647598437864200465143322

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