Properties

Label 2-9576-1.1-c1-0-3
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  − 2.53·5-s + 7-s − 4.91·11-s − 0.165·13-s − 6.84·17-s + 19-s − 2.16·23-s + 1.43·25-s − 4.70·29-s − 7.07·31-s − 2.53·35-s − 0.746·37-s − 3.07·41-s − 3.88·47-s + 49-s + 0.702·53-s + 12.4·55-s + 13.3·59-s + 6·61-s + 0.419·65-s − 13.0·67-s + 8.26·71-s − 5.05·73-s − 4.91·77-s + 9.22·79-s − 8.19·83-s + 17.3·85-s + ⋯
L(s)  = 1  − 1.13·5-s + 0.377·7-s − 1.48·11-s − 0.0459·13-s − 1.66·17-s + 0.229·19-s − 0.451·23-s + 0.287·25-s − 0.873·29-s − 1.27·31-s − 0.428·35-s − 0.122·37-s − 0.480·41-s − 0.566·47-s + 0.142·49-s + 0.0964·53-s + 1.68·55-s + 1.73·59-s + 0.768·61-s + 0.0520·65-s − 1.59·67-s + 0.981·71-s − 0.591·73-s − 0.559·77-s + 1.03·79-s − 0.899·83-s + 1.88·85-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: $\chi_{9576} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 9576,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4880124089\)
\(L(\frac12)\) \(\approx\) \(0.4880124089\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.53T + 5T^{2} \)
11 \( 1 + 4.91T + 11T^{2} \)
13 \( 1 + 0.165T + 13T^{2} \)
17 \( 1 + 6.84T + 17T^{2} \)
23 \( 1 + 2.16T + 23T^{2} \)
29 \( 1 + 4.70T + 29T^{2} \)
31 \( 1 + 7.07T + 31T^{2} \)
37 \( 1 + 0.746T + 37T^{2} \)
41 \( 1 + 3.07T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 3.88T + 47T^{2} \)
53 \( 1 - 0.702T + 53T^{2} \)
59 \( 1 - 13.3T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 + 13.0T + 67T^{2} \)
71 \( 1 - 8.26T + 71T^{2} \)
73 \( 1 + 5.05T + 73T^{2} \)
79 \( 1 - 9.22T + 79T^{2} \)
83 \( 1 + 8.19T + 83T^{2} \)
89 \( 1 - 9.38T + 89T^{2} \)
97 \( 1 + 3.65T + 97T^{2} \)
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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

−7.56574373467446865642442213231, −7.30529304155128337310313550832, −6.40945460484058457153849874702, −5.45272280538112849703193518582, −4.95106684766348840680704185229, −4.14236618183931777645538345078, −3.57501215913349413776621631070, −2.56399640259234639467548118934, −1.86314391387657025371408245253, −0.31422576490895026381145820510, 0.31422576490895026381145820510, 1.86314391387657025371408245253, 2.56399640259234639467548118934, 3.57501215913349413776621631070, 4.14236618183931777645538345078, 4.95106684766348840680704185229, 5.45272280538112849703193518582, 6.40945460484058457153849874702, 7.30529304155128337310313550832, 7.56574373467446865642442213231

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