Properties

Label 2-9576-1.1-c1-0-100
Degree $2$
Conductor $9576$
Sign $-1$
Analytic cond. $76.4647$
Root an. cond. $8.74441$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·5-s + 7-s + 6·11-s − 4·17-s − 19-s + 4·23-s − 25-s − 2·29-s − 2·31-s − 2·35-s − 10·41-s − 4·43-s + 4·47-s + 49-s − 14·53-s − 12·55-s − 2·61-s + 12·67-s − 2·73-s + 6·77-s + 10·79-s − 6·83-s + 8·85-s − 10·89-s + 2·95-s − 2·97-s − 6·101-s + ⋯
L(s)  = 1  − 0.894·5-s + 0.377·7-s + 1.80·11-s − 0.970·17-s − 0.229·19-s + 0.834·23-s − 1/5·25-s − 0.371·29-s − 0.359·31-s − 0.338·35-s − 1.56·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s − 1.92·53-s − 1.61·55-s − 0.256·61-s + 1.46·67-s − 0.234·73-s + 0.683·77-s + 1.12·79-s − 0.658·83-s + 0.867·85-s − 1.05·89-s + 0.205·95-s − 0.203·97-s − 0.597·101-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: yes
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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 14 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 - 10 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−7.22691893437655949048325903421, −6.76790358391537788637589702705, −6.16470799052164746476181404201, −5.13367461976556064774893175522, −4.45481007992373788200109568817, −3.85516632097326418693035259867, −3.26413020570320937120106891489, −2.03516272748559446734667846717, −1.24913849907889685446558554611, 0, 1.24913849907889685446558554611, 2.03516272748559446734667846717, 3.26413020570320937120106891489, 3.85516632097326418693035259867, 4.45481007992373788200109568817, 5.13367461976556064774893175522, 6.16470799052164746476181404201, 6.76790358391537788637589702705, 7.22691893437655949048325903421

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