Properties

Label 2-9576-1.1-c1-0-1
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  − 3.96·5-s − 7-s − 3.97·11-s − 0.477·13-s + 0.685·17-s + 19-s + 2.47·23-s + 10.6·25-s − 9.12·29-s − 2.68·31-s + 3.96·35-s − 2.95·37-s − 8.20·41-s − 11.3·43-s − 3.73·47-s + 49-s + 4.87·53-s + 15.7·55-s − 7.33·59-s − 5.18·61-s + 1.89·65-s − 10.1·67-s + 10.1·71-s − 13.2·73-s + 3.97·77-s − 2.48·79-s + 7.20·83-s + ⋯
L(s)  = 1  − 1.77·5-s − 0.377·7-s − 1.19·11-s − 0.132·13-s + 0.166·17-s + 0.229·19-s + 0.516·23-s + 2.13·25-s − 1.69·29-s − 0.482·31-s + 0.669·35-s − 0.486·37-s − 1.28·41-s − 1.73·43-s − 0.545·47-s + 0.142·49-s + 0.669·53-s + 2.12·55-s − 0.954·59-s − 0.663·61-s + 0.234·65-s − 1.23·67-s + 1.20·71-s − 1.54·73-s + 0.453·77-s − 0.279·79-s + 0.790·83-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

\(L(1)\) \(\approx\) \(0.2640886410\)
\(L(\frac12)\) \(\approx\) \(0.2640886410\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.96T + 5T^{2} \)
11 \( 1 + 3.97T + 11T^{2} \)
13 \( 1 + 0.477T + 13T^{2} \)
17 \( 1 - 0.685T + 17T^{2} \)
23 \( 1 - 2.47T + 23T^{2} \)
29 \( 1 + 9.12T + 29T^{2} \)
31 \( 1 + 2.68T + 31T^{2} \)
37 \( 1 + 2.95T + 37T^{2} \)
41 \( 1 + 8.20T + 41T^{2} \)
43 \( 1 + 11.3T + 43T^{2} \)
47 \( 1 + 3.73T + 47T^{2} \)
53 \( 1 - 4.87T + 53T^{2} \)
59 \( 1 + 7.33T + 59T^{2} \)
61 \( 1 + 5.18T + 61T^{2} \)
67 \( 1 + 10.1T + 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 + 13.2T + 73T^{2} \)
79 \( 1 + 2.48T + 79T^{2} \)
83 \( 1 - 7.20T + 83T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 - 16.6T + 97T^{2} \)
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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

−7.56737772544516502915428972140, −7.28644715256833706394761359743, −6.50710856148276435699217786975, −5.41529317268082019784555513309, −4.96598751433253753243715901042, −4.09991875240501632655732017756, −3.37705871639290510775513023329, −2.95169659841423967654242552819, −1.67110364763798232667035285811, −0.23828535909734595305214095949, 0.23828535909734595305214095949, 1.67110364763798232667035285811, 2.95169659841423967654242552819, 3.37705871639290510775513023329, 4.09991875240501632655732017756, 4.96598751433253753243715901042, 5.41529317268082019784555513309, 6.50710856148276435699217786975, 7.28644715256833706394761359743, 7.56737772544516502915428972140

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