Properties

Label 2-9576-1.1-c1-0-33
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  + 1.79·5-s + 7-s − 0.736·11-s − 5.06·13-s + 3.33·17-s + 19-s − 7.06·23-s − 1.79·25-s − 5.27·29-s − 7.80·31-s + 1.79·35-s + 8.33·37-s + 5.58·41-s + 7.84·47-s + 49-s + 1.27·53-s − 1.31·55-s − 6.90·59-s + 6·61-s − 9.08·65-s + 14.7·67-s + 12.0·71-s + 9.87·73-s − 0.736·77-s − 0.805·79-s + 9.38·83-s + 5.97·85-s + ⋯
L(s)  = 1  + 0.801·5-s + 0.377·7-s − 0.222·11-s − 1.40·13-s + 0.808·17-s + 0.229·19-s − 1.47·23-s − 0.358·25-s − 0.979·29-s − 1.40·31-s + 0.302·35-s + 1.36·37-s + 0.871·41-s + 1.14·47-s + 0.142·49-s + 0.175·53-s − 0.177·55-s − 0.898·59-s + 0.768·61-s − 1.12·65-s + 1.79·67-s + 1.43·71-s + 1.15·73-s − 0.0839·77-s − 0.0906·79-s + 1.03·83-s + 0.647·85-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

\(L(1)\) \(\approx\) \(2.140532371\)
\(L(\frac12)\) \(\approx\) \(2.140532371\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.79T + 5T^{2} \)
11 \( 1 + 0.736T + 11T^{2} \)
13 \( 1 + 5.06T + 13T^{2} \)
17 \( 1 - 3.33T + 17T^{2} \)
23 \( 1 + 7.06T + 23T^{2} \)
29 \( 1 + 5.27T + 29T^{2} \)
31 \( 1 + 7.80T + 31T^{2} \)
37 \( 1 - 8.33T + 37T^{2} \)
41 \( 1 - 5.58T + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 7.84T + 47T^{2} \)
53 \( 1 - 1.27T + 53T^{2} \)
59 \( 1 + 6.90T + 59T^{2} \)
61 \( 1 - 6T + 61T^{2} \)
67 \( 1 - 14.7T + 67T^{2} \)
71 \( 1 - 12.0T + 71T^{2} \)
73 \( 1 - 9.87T + 73T^{2} \)
79 \( 1 + 0.805T + 79T^{2} \)
83 \( 1 - 9.38T + 83T^{2} \)
89 \( 1 - 4.26T + 89T^{2} \)
97 \( 1 - 18.9T + 97T^{2} \)
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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

−7.69483541606010408088783795773, −7.16651290043403391744809839586, −6.14511356796167422456840516027, −5.61510946919811668310432851036, −5.11704527273308374420525792952, −4.21071162778156798931014871677, −3.46059325924665562834732104434, −2.23198625118616367925005824542, −2.07004711388864479692031643526, −0.67654009814019598512140254832, 0.67654009814019598512140254832, 2.07004711388864479692031643526, 2.23198625118616367925005824542, 3.46059325924665562834732104434, 4.21071162778156798931014871677, 5.11704527273308374420525792952, 5.61510946919811668310432851036, 6.14511356796167422456840516027, 7.16651290043403391744809839586, 7.69483541606010408088783795773

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