Properties

Label 2-9576-1.1-c1-0-113
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.23·5-s + 7-s − 2.85·11-s − 1.76·13-s − 0.854·17-s + 19-s − 6.23·23-s + 4.38·29-s − 2.38·31-s + 2.23·35-s + 1.47·37-s + 10.5·41-s − 6.94·43-s − 7·47-s + 49-s + 1.85·53-s − 6.38·55-s + 10.7·59-s − 12.7·61-s − 3.94·65-s − 5.61·67-s − 3.94·71-s − 0.145·73-s − 2.85·77-s − 4.47·79-s + 11.3·83-s − 1.90·85-s + ⋯
L(s)  = 1  + 0.999·5-s + 0.377·7-s − 0.860·11-s − 0.489·13-s − 0.207·17-s + 0.229·19-s − 1.30·23-s + 0.813·29-s − 0.427·31-s + 0.377·35-s + 0.242·37-s + 1.64·41-s − 1.05·43-s − 1.02·47-s + 0.142·49-s + 0.254·53-s − 0.860·55-s + 1.39·59-s − 1.62·61-s − 0.489·65-s − 0.686·67-s − 0.468·71-s − 0.0170·73-s − 0.325·77-s − 0.503·79-s + 1.24·83-s − 0.207·85-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 9576,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23T + 5T^{2} \)
11 \( 1 + 2.85T + 11T^{2} \)
13 \( 1 + 1.76T + 13T^{2} \)
17 \( 1 + 0.854T + 17T^{2} \)
23 \( 1 + 6.23T + 23T^{2} \)
29 \( 1 - 4.38T + 29T^{2} \)
31 \( 1 + 2.38T + 31T^{2} \)
37 \( 1 - 1.47T + 37T^{2} \)
41 \( 1 - 10.5T + 41T^{2} \)
43 \( 1 + 6.94T + 43T^{2} \)
47 \( 1 + 7T + 47T^{2} \)
53 \( 1 - 1.85T + 53T^{2} \)
59 \( 1 - 10.7T + 59T^{2} \)
61 \( 1 + 12.7T + 61T^{2} \)
67 \( 1 + 5.61T + 67T^{2} \)
71 \( 1 + 3.94T + 71T^{2} \)
73 \( 1 + 0.145T + 73T^{2} \)
79 \( 1 + 4.47T + 79T^{2} \)
83 \( 1 - 11.3T + 83T^{2} \)
89 \( 1 + 0.763T + 89T^{2} \)
97 \( 1 + 3.76T + 97T^{2} \)
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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

−7.49518463384154262959322128698, −6.52492701833764503946545740076, −5.96909335183284681638111578499, −5.31188770468733093912461425437, −4.72229861920770328497778172301, −3.87566691652684810545411004986, −2.77597498803856012760135702109, −2.22674714033674305834178525694, −1.38946216191953009008290482688, 0, 1.38946216191953009008290482688, 2.22674714033674305834178525694, 2.77597498803856012760135702109, 3.87566691652684810545411004986, 4.72229861920770328497778172301, 5.31188770468733093912461425437, 5.96909335183284681638111578499, 6.52492701833764503946545740076, 7.49518463384154262959322128698

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