Properties

Label 2-9576-1.1-c1-0-52
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  − 3.61·5-s + 7-s + 5.85·11-s + 5.23·13-s + 2.47·17-s − 19-s + 5.70·23-s + 8.09·25-s + 0.854·29-s + 4.47·31-s − 3.61·35-s − 0.0901·37-s + 6.09·41-s + 6.85·43-s + 11.8·47-s + 49-s + 3.38·53-s − 21.1·55-s − 4.09·59-s − 7.85·61-s − 18.9·65-s − 14.9·67-s − 13.5·71-s + 8.76·73-s + 5.85·77-s − 6.85·79-s − 4·83-s + ⋯
L(s)  = 1  − 1.61·5-s + 0.377·7-s + 1.76·11-s + 1.45·13-s + 0.599·17-s − 0.229·19-s + 1.19·23-s + 1.61·25-s + 0.158·29-s + 0.803·31-s − 0.611·35-s − 0.0148·37-s + 0.951·41-s + 1.04·43-s + 1.72·47-s + 0.142·49-s + 0.464·53-s − 2.85·55-s − 0.532·59-s − 1.00·61-s − 2.34·65-s − 1.82·67-s − 1.60·71-s + 1.02·73-s + 0.667·77-s − 0.771·79-s − 0.439·83-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

\(L(1)\) \(\approx\) \(2.242996608\)
\(L(\frac12)\) \(\approx\) \(2.242996608\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.61T + 5T^{2} \)
11 \( 1 - 5.85T + 11T^{2} \)
13 \( 1 - 5.23T + 13T^{2} \)
17 \( 1 - 2.47T + 17T^{2} \)
23 \( 1 - 5.70T + 23T^{2} \)
29 \( 1 - 0.854T + 29T^{2} \)
31 \( 1 - 4.47T + 31T^{2} \)
37 \( 1 + 0.0901T + 37T^{2} \)
41 \( 1 - 6.09T + 41T^{2} \)
43 \( 1 - 6.85T + 43T^{2} \)
47 \( 1 - 11.8T + 47T^{2} \)
53 \( 1 - 3.38T + 53T^{2} \)
59 \( 1 + 4.09T + 59T^{2} \)
61 \( 1 + 7.85T + 61T^{2} \)
67 \( 1 + 14.9T + 67T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 - 8.76T + 73T^{2} \)
79 \( 1 + 6.85T + 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + 1.09T + 89T^{2} \)
97 \( 1 + 3.56T + 97T^{2} \)
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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

−7.56227729721006237410556702592, −7.21770093575038299974209531980, −6.33167366566478104734830640402, −5.80406933911011178746656381321, −4.56491387738026598295641692599, −4.18608526971003170217114372318, −3.59643036998779116244082795060, −2.87552512628305221414259723137, −1.34482670994166467909836355691, −0.848867351532436814216510324145, 0.848867351532436814216510324145, 1.34482670994166467909836355691, 2.87552512628305221414259723137, 3.59643036998779116244082795060, 4.18608526971003170217114372318, 4.56491387738026598295641692599, 5.80406933911011178746656381321, 6.33167366566478104734830640402, 7.21770093575038299974209531980, 7.56227729721006237410556702592

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