Properties

Label 2-9576-1.1-c1-0-64
Degree $2$
Conductor $9576$
Sign $1$
Analytic cond. $76.4647$
Root an. cond. $8.74441$
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  + 2.73·5-s − 7-s + 3.46·11-s + 4·13-s + 6.73·17-s + 19-s − 7.46·23-s + 2.46·25-s + 0.732·29-s + 3.46·31-s − 2.73·35-s − 0.535·37-s + 4.53·41-s + 4.92·43-s − 11.6·47-s + 49-s − 2.19·53-s + 9.46·55-s + 8·59-s + 11.4·61-s + 10.9·65-s − 4·67-s + 4.19·71-s + 7.46·73-s − 3.46·77-s + 1.46·79-s + 10.1·83-s + ⋯
L(s)  = 1  + 1.22·5-s − 0.377·7-s + 1.04·11-s + 1.10·13-s + 1.63·17-s + 0.229·19-s − 1.55·23-s + 0.492·25-s + 0.135·29-s + 0.622·31-s − 0.461·35-s − 0.0881·37-s + 0.708·41-s + 0.751·43-s − 1.70·47-s + 0.142·49-s − 0.301·53-s + 1.27·55-s + 1.04·59-s + 1.46·61-s + 1.35·65-s − 0.488·67-s + 0.497·71-s + 0.873·73-s − 0.394·77-s + 0.164·79-s + 1.11·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(9576\)    =    \(2^{3} \cdot 3^{2} \cdot 7 \cdot 19\)
Sign: $1$
Analytic conductor: \(76.4647\)
Root analytic conductor: \(8.74441\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9576,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.309135891\)
\(L(\frac12)\) \(\approx\) \(3.309135891\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 + T \)
19 \( 1 - T \)
good5 \( 1 - 2.73T + 5T^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 - 6.73T + 17T^{2} \)
23 \( 1 + 7.46T + 23T^{2} \)
29 \( 1 - 0.732T + 29T^{2} \)
31 \( 1 - 3.46T + 31T^{2} \)
37 \( 1 + 0.535T + 37T^{2} \)
41 \( 1 - 4.53T + 41T^{2} \)
43 \( 1 - 4.92T + 43T^{2} \)
47 \( 1 + 11.6T + 47T^{2} \)
53 \( 1 + 2.19T + 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 11.4T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 4.19T + 71T^{2} \)
73 \( 1 - 7.46T + 73T^{2} \)
79 \( 1 - 1.46T + 79T^{2} \)
83 \( 1 - 10.1T + 83T^{2} \)
89 \( 1 - 10.3T + 89T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−7.82508858757232868766556216147, −6.72033452185198052128553924444, −6.28104305201082942431105533896, −5.78211797709902953350540770050, −5.15437858554929948956192632811, −3.97313550308581969774860911952, −3.55399166021064744249868098288, −2.53939371161969454489779208379, −1.62569836511552949477421568566, −0.950352055993476671989546866435, 0.950352055993476671989546866435, 1.62569836511552949477421568566, 2.53939371161969454489779208379, 3.55399166021064744249868098288, 3.97313550308581969774860911952, 5.15437858554929948956192632811, 5.78211797709902953350540770050, 6.28104305201082942431105533896, 6.72033452185198052128553924444, 7.82508858757232868766556216147

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