Properties

Label 2-9576-1.1-c1-0-19
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  − 0.732·5-s − 7-s − 3.46·11-s + 4·13-s + 3.26·17-s + 19-s − 0.535·23-s − 4.46·25-s − 2.73·29-s − 3.46·31-s + 0.732·35-s − 7.46·37-s + 11.4·41-s − 8.92·43-s + 5.66·47-s + 49-s + 8.19·53-s + 2.53·55-s + 8·59-s + 4.53·61-s − 2.92·65-s − 4·67-s − 6.19·71-s + 0.535·73-s + 3.46·77-s − 5.46·79-s − 0.196·83-s + ⋯
L(s)  = 1  − 0.327·5-s − 0.377·7-s − 1.04·11-s + 1.10·13-s + 0.792·17-s + 0.229·19-s − 0.111·23-s − 0.892·25-s − 0.507·29-s − 0.622·31-s + 0.123·35-s − 1.22·37-s + 1.79·41-s − 1.36·43-s + 0.825·47-s + 0.142·49-s + 1.12·53-s + 0.341·55-s + 1.04·59-s + 0.580·61-s − 0.363·65-s − 0.488·67-s − 0.735·71-s + 0.0627·73-s + 0.394·77-s − 0.614·79-s − 0.0215·83-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

\(L(1)\) \(\approx\) \(1.466552494\)
\(L(\frac12)\) \(\approx\) \(1.466552494\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.732T + 5T^{2} \)
11 \( 1 + 3.46T + 11T^{2} \)
13 \( 1 - 4T + 13T^{2} \)
17 \( 1 - 3.26T + 17T^{2} \)
23 \( 1 + 0.535T + 23T^{2} \)
29 \( 1 + 2.73T + 29T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
37 \( 1 + 7.46T + 37T^{2} \)
41 \( 1 - 11.4T + 41T^{2} \)
43 \( 1 + 8.92T + 43T^{2} \)
47 \( 1 - 5.66T + 47T^{2} \)
53 \( 1 - 8.19T + 53T^{2} \)
59 \( 1 - 8T + 59T^{2} \)
61 \( 1 - 4.53T + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + 6.19T + 71T^{2} \)
73 \( 1 - 0.535T + 73T^{2} \)
79 \( 1 + 5.46T + 79T^{2} \)
83 \( 1 + 0.196T + 83T^{2} \)
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

−7.48966471687452736005384545564, −7.30300186993056739819950634438, −6.16354047907991182134082748996, −5.69475418493493397485105148642, −5.06474179561794531607418513894, −3.96967273762296694840005577259, −3.55163173685440234926923557660, −2.68504311789875791285292515414, −1.71977579391393835244646285291, −0.57458679088412248307946885761, 0.57458679088412248307946885761, 1.71977579391393835244646285291, 2.68504311789875791285292515414, 3.55163173685440234926923557660, 3.96967273762296694840005577259, 5.06474179561794531607418513894, 5.69475418493493397485105148642, 6.16354047907991182134082748996, 7.30300186993056739819950634438, 7.48966471687452736005384545564

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