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 − 3.12·5-s + 4.33·7-s + 9-s − 3.88·11-s − 5.76·13-s + 3.12·15-s + 1.77·17-s + 1.14·19-s − 4.33·21-s + 23-s + 4.76·25-s − 27-s − 29-s + 5.76·31-s + 3.88·33-s − 13.5·35-s + 9.00·37-s + 5.76·39-s − 1.21·41-s + 6.15·43-s − 3.12·45-s − 1.33·47-s + 11.7·49-s − 1.77·51-s − 5.98·53-s + 12.1·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.39·5-s + 1.63·7-s + 0.333·9-s − 1.17·11-s − 1.60·13-s + 0.806·15-s + 0.431·17-s + 0.261·19-s − 0.946·21-s + 0.208·23-s + 0.953·25-s − 0.192·27-s − 0.185·29-s + 1.03·31-s + 0.676·33-s − 2.28·35-s + 1.48·37-s + 0.923·39-s − 0.190·41-s + 0.939·43-s − 0.465·45-s − 0.194·47-s + 1.68·49-s − 0.248·51-s − 0.821·53-s + 1.63·55-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 \( 1 + 3.12T + 5T^{2} \)
7 \( 1 - 4.33T + 7T^{2} \)
11 \( 1 + 3.88T + 11T^{2} \)
13 \( 1 + 5.76T + 13T^{2} \)
17 \( 1 - 1.77T + 17T^{2} \)
19 \( 1 - 1.14T + 19T^{2} \)
31 \( 1 - 5.76T + 31T^{2} \)
37 \( 1 - 9.00T + 37T^{2} \)
41 \( 1 + 1.21T + 41T^{2} \)
43 \( 1 - 6.15T + 43T^{2} \)
47 \( 1 + 1.33T + 47T^{2} \)
53 \( 1 + 5.98T + 53T^{2} \)
59 \( 1 + 7.53T + 59T^{2} \)
61 \( 1 - 2.35T + 61T^{2} \)
67 \( 1 + 3.03T + 67T^{2} \)
71 \( 1 - 6.64T + 71T^{2} \)
73 \( 1 - 2.08T + 73T^{2} \)
79 \( 1 + 13.0T + 79T^{2} \)
83 \( 1 + 14.3T + 83T^{2} \)
89 \( 1 + 4.89T + 89T^{2} \)
97 \( 1 - 7.62T + 97T^{2} \)
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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

−7.61073105107571121973592384811, −7.16891599877624844883765428032, −5.95409144507712332215833800448, −5.14375755872672127924746698434, −4.69218425331937461505389598792, −4.26858007816131703859021471600, −3.05533703408507025714663261569, −2.26523177655415254199657651043, −1.04022949183673160588447109748, 0, 1.04022949183673160588447109748, 2.26523177655415254199657651043, 3.05533703408507025714663261569, 4.26858007816131703859021471600, 4.69218425331937461505389598792, 5.14375755872672127924746698434, 5.95409144507712332215833800448, 7.16891599877624844883765428032, 7.61073105107571121973592384811

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