Properties

Label 2-9576-1.1-c1-0-117
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  + 1.64·5-s + 7-s − 2·11-s − 4·13-s + 5.64·17-s + 19-s + 6·23-s − 2.29·25-s − 0.354·29-s − 9.29·31-s + 1.64·35-s − 5.29·37-s − 12.5·41-s − 2·43-s − 7.64·47-s + 49-s − 6.93·53-s − 3.29·55-s + 6.58·59-s + 13.2·61-s − 6.58·65-s − 6.58·67-s − 5.64·71-s + 1.29·73-s − 2·77-s + 0.708·79-s − 1.06·83-s + ⋯
L(s)  = 1  + 0.736·5-s + 0.377·7-s − 0.603·11-s − 1.10·13-s + 1.36·17-s + 0.229·19-s + 1.25·23-s − 0.458·25-s − 0.0657·29-s − 1.66·31-s + 0.278·35-s − 0.869·37-s − 1.96·41-s − 0.304·43-s − 1.11·47-s + 0.142·49-s − 0.952·53-s − 0.443·55-s + 0.857·59-s + 1.70·61-s − 0.816·65-s − 0.804·67-s − 0.670·71-s + 0.151·73-s − 0.227·77-s + 0.0797·79-s − 0.116·83-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 \( 1 - 1.64T + 5T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 4T + 13T^{2} \)
17 \( 1 - 5.64T + 17T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + 0.354T + 29T^{2} \)
31 \( 1 + 9.29T + 31T^{2} \)
37 \( 1 + 5.29T + 37T^{2} \)
41 \( 1 + 12.5T + 41T^{2} \)
43 \( 1 + 2T + 43T^{2} \)
47 \( 1 + 7.64T + 47T^{2} \)
53 \( 1 + 6.93T + 53T^{2} \)
59 \( 1 - 6.58T + 59T^{2} \)
61 \( 1 - 13.2T + 61T^{2} \)
67 \( 1 + 6.58T + 67T^{2} \)
71 \( 1 + 5.64T + 71T^{2} \)
73 \( 1 - 1.29T + 73T^{2} \)
79 \( 1 - 0.708T + 79T^{2} \)
83 \( 1 + 1.06T + 83T^{2} \)
89 \( 1 + 6T + 89T^{2} \)
97 \( 1 + 4.58T + 97T^{2} \)
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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

−7.31421030069009047839379644265, −6.81966215007009031488342702908, −5.78818050625183393982499936125, −5.16583143215371451030444266880, −4.98057801195994588005802727173, −3.65976875108184288472763538888, −3.02901787762875279513982962939, −2.08926892646436461320551620342, −1.39919491925572497014274298275, 0, 1.39919491925572497014274298275, 2.08926892646436461320551620342, 3.02901787762875279513982962939, 3.65976875108184288472763538888, 4.98057801195994588005802727173, 5.16583143215371451030444266880, 5.78818050625183393982499936125, 6.81966215007009031488342702908, 7.31421030069009047839379644265

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