Properties

Label 2-9576-1.1-c1-0-76
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  + 3.88·5-s + 7-s − 3.21·11-s + 5.39·13-s + 4.41·17-s + 19-s + 3.39·23-s + 10.1·25-s + 7.28·29-s + 0.187·31-s + 3.88·35-s − 4.60·37-s + 9.77·41-s − 4.45·47-s + 49-s − 11.2·53-s − 12.4·55-s + 5.77·59-s + 6·61-s + 20.9·65-s + 2.48·67-s − 12.4·71-s − 4.08·73-s − 3.21·77-s + 2.68·79-s − 3.92·83-s + 17.1·85-s + ⋯
L(s)  = 1  + 1.73·5-s + 0.377·7-s − 0.968·11-s + 1.49·13-s + 1.06·17-s + 0.229·19-s + 0.708·23-s + 2.02·25-s + 1.35·29-s + 0.0336·31-s + 0.657·35-s − 0.757·37-s + 1.52·41-s − 0.649·47-s + 0.142·49-s − 1.55·53-s − 1.68·55-s + 0.751·59-s + 0.768·61-s + 2.60·65-s + 0.303·67-s − 1.47·71-s − 0.478·73-s − 0.365·77-s + 0.302·79-s − 0.431·83-s + 1.86·85-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

\(L(1)\) \(\approx\) \(3.789077635\)
\(L(\frac12)\) \(\approx\) \(3.789077635\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.88T + 5T^{2} \)
11 \( 1 + 3.21T + 11T^{2} \)
13 \( 1 - 5.39T + 13T^{2} \)
17 \( 1 - 4.41T + 17T^{2} \)
23 \( 1 - 3.39T + 23T^{2} \)
29 \( 1 - 7.28T + 29T^{2} \)
31 \( 1 - 0.187T + 31T^{2} \)
37 \( 1 + 4.60T + 37T^{2} \)
41 \( 1 - 9.77T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + 4.45T + 47T^{2} \)
53 \( 1 + 11.2T + 53T^{2} \)
59 \( 1 - 5.77T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 2.48T + 67T^{2} \)
71 \( 1 + 12.4T + 71T^{2} \)
73 \( 1 + 4.08T + 73T^{2} \)
79 \( 1 - 2.68T + 79T^{2} \)
83 \( 1 + 3.92T + 83T^{2} \)
89 \( 1 + 2.70T + 89T^{2} \)
97 \( 1 + 0.227T + 97T^{2} \)
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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

−7.79999568231853951501627597291, −6.81317202076819011414484188184, −6.26239086546832364142451933834, −5.50614276935348184412470375611, −5.28587843614864286461154603084, −4.31200151388813041918371951764, −3.16082321397427373006292418074, −2.64806692057089703019525312092, −1.59944121109028231427853471114, −1.04098793486253116509606413594, 1.04098793486253116509606413594, 1.59944121109028231427853471114, 2.64806692057089703019525312092, 3.16082321397427373006292418074, 4.31200151388813041918371951764, 5.28587843614864286461154603084, 5.50614276935348184412470375611, 6.26239086546832364142451933834, 6.81317202076819011414484188184, 7.79999568231853951501627597291

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