Properties

Degree $2$
Conductor $8004$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 1.34·5-s + 2.91·7-s + 9-s + 3.69·11-s − 6.03·13-s − 1.34·15-s + 5.75·17-s − 6.30·19-s − 2.91·21-s − 23-s − 3.18·25-s − 27-s + 29-s − 0.671·31-s − 3.69·33-s + 3.93·35-s − 4.42·37-s + 6.03·39-s − 2.20·41-s − 8.58·43-s + 1.34·45-s + 0.549·47-s + 1.51·49-s − 5.75·51-s − 2.93·53-s + 4.98·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.602·5-s + 1.10·7-s + 0.333·9-s + 1.11·11-s − 1.67·13-s − 0.347·15-s + 1.39·17-s − 1.44·19-s − 0.636·21-s − 0.208·23-s − 0.636·25-s − 0.192·27-s + 0.185·29-s − 0.120·31-s − 0.643·33-s + 0.664·35-s − 0.728·37-s + 0.966·39-s − 0.343·41-s − 1.30·43-s + 0.200·45-s + 0.0801·47-s + 0.216·49-s − 0.805·51-s − 0.403·53-s + 0.671·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8004\)    =    \(2^{2} \cdot 3 \cdot 23 \cdot 29\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{8004} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8004,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 - T \)
good5 \( 1 - 1.34T + 5T^{2} \)
7 \( 1 - 2.91T + 7T^{2} \)
11 \( 1 - 3.69T + 11T^{2} \)
13 \( 1 + 6.03T + 13T^{2} \)
17 \( 1 - 5.75T + 17T^{2} \)
19 \( 1 + 6.30T + 19T^{2} \)
31 \( 1 + 0.671T + 31T^{2} \)
37 \( 1 + 4.42T + 37T^{2} \)
41 \( 1 + 2.20T + 41T^{2} \)
43 \( 1 + 8.58T + 43T^{2} \)
47 \( 1 - 0.549T + 47T^{2} \)
53 \( 1 + 2.93T + 53T^{2} \)
59 \( 1 + 4.20T + 59T^{2} \)
61 \( 1 + 10.6T + 61T^{2} \)
67 \( 1 + 3.98T + 67T^{2} \)
71 \( 1 + 9.08T + 71T^{2} \)
73 \( 1 + 2.11T + 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 + 6.50T + 83T^{2} \)
89 \( 1 - 4.64T + 89T^{2} \)
97 \( 1 + 1.50T + 97T^{2} \)
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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

−7.47125150363600545128312583852, −6.75428657448052401120738669022, −6.04878013031020244934094896777, −5.38753644062001442255935172663, −4.72608812921343221071790271003, −4.17119822480674882545159977542, −3.05498929682246183837394596204, −1.89625518394172723780987431180, −1.49253465800154112722354071227, 0, 1.49253465800154112722354071227, 1.89625518394172723780987431180, 3.05498929682246183837394596204, 4.17119822480674882545159977542, 4.72608812921343221071790271003, 5.38753644062001442255935172663, 6.04878013031020244934094896777, 6.75428657448052401120738669022, 7.47125150363600545128312583852

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