Properties

Label 2-9576-1.1-c1-0-93
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  − 1.23·5-s − 7-s + 4.47·11-s − 4.47·13-s + 1.23·17-s + 19-s + 0.472·23-s − 3.47·25-s − 7.23·29-s + 6.47·31-s + 1.23·35-s − 4.47·37-s + 2·41-s + 8·43-s + 3.23·47-s + 49-s + 3.23·53-s − 5.52·55-s + 4·59-s − 3.52·61-s + 5.52·65-s − 1.52·67-s − 5.23·71-s + 4.47·73-s − 4.47·77-s − 12·79-s + 5.70·83-s + ⋯
L(s)  = 1  − 0.552·5-s − 0.377·7-s + 1.34·11-s − 1.24·13-s + 0.299·17-s + 0.229·19-s + 0.0984·23-s − 0.694·25-s − 1.34·29-s + 1.16·31-s + 0.208·35-s − 0.735·37-s + 0.312·41-s + 1.21·43-s + 0.472·47-s + 0.142·49-s + 0.444·53-s − 0.745·55-s + 0.520·59-s − 0.451·61-s + 0.685·65-s − 0.186·67-s − 0.621·71-s + 0.523·73-s − 0.509·77-s − 1.35·79-s + 0.626·83-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: \(1\)
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 \( 1 + 1.23T + 5T^{2} \)
11 \( 1 - 4.47T + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 - 1.23T + 17T^{2} \)
23 \( 1 - 0.472T + 23T^{2} \)
29 \( 1 + 7.23T + 29T^{2} \)
31 \( 1 - 6.47T + 31T^{2} \)
37 \( 1 + 4.47T + 37T^{2} \)
41 \( 1 - 2T + 41T^{2} \)
43 \( 1 - 8T + 43T^{2} \)
47 \( 1 - 3.23T + 47T^{2} \)
53 \( 1 - 3.23T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 3.52T + 61T^{2} \)
67 \( 1 + 1.52T + 67T^{2} \)
71 \( 1 + 5.23T + 71T^{2} \)
73 \( 1 - 4.47T + 73T^{2} \)
79 \( 1 + 12T + 79T^{2} \)
83 \( 1 - 5.70T + 83T^{2} \)
89 \( 1 + 10.9T + 89T^{2} \)
97 \( 1 - 3.52T + 97T^{2} \)
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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

−7.31632370874229010607916925332, −6.81734263652294456454082607283, −5.98090463534957735751113656080, −5.34994363652599251282863156397, −4.35195028037816390684284027779, −3.93676998670209969985001390485, −3.09453400936313081601225809344, −2.21713228050150796281853352126, −1.15104211008351401609105827360, 0, 1.15104211008351401609105827360, 2.21713228050150796281853352126, 3.09453400936313081601225809344, 3.93676998670209969985001390485, 4.35195028037816390684284027779, 5.34994363652599251282863156397, 5.98090463534957735751113656080, 6.81734263652294456454082607283, 7.31632370874229010607916925332

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