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 + 0.461·5-s + 0.732·7-s + 9-s + 0.317·11-s + 1.73·13-s − 0.461·15-s − 1.85·17-s − 0.400·19-s − 0.732·21-s − 23-s − 4.78·25-s − 27-s + 29-s − 2.76·31-s − 0.317·33-s + 0.338·35-s + 6.86·37-s − 1.73·39-s + 2.88·41-s + 7.54·43-s + 0.461·45-s − 11.3·47-s − 6.46·49-s + 1.85·51-s − 10.8·53-s + 0.146·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.206·5-s + 0.276·7-s + 0.333·9-s + 0.0955·11-s + 0.480·13-s − 0.119·15-s − 0.450·17-s − 0.0919·19-s − 0.159·21-s − 0.208·23-s − 0.957·25-s − 0.192·27-s + 0.185·29-s − 0.496·31-s − 0.0551·33-s + 0.0571·35-s + 1.12·37-s − 0.277·39-s + 0.450·41-s + 1.15·43-s + 0.0688·45-s − 1.66·47-s − 0.923·49-s + 0.260·51-s − 1.48·53-s + 0.0197·55-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 \( 1 - 0.461T + 5T^{2} \)
7 \( 1 - 0.732T + 7T^{2} \)
11 \( 1 - 0.317T + 11T^{2} \)
13 \( 1 - 1.73T + 13T^{2} \)
17 \( 1 + 1.85T + 17T^{2} \)
19 \( 1 + 0.400T + 19T^{2} \)
31 \( 1 + 2.76T + 31T^{2} \)
37 \( 1 - 6.86T + 37T^{2} \)
41 \( 1 - 2.88T + 41T^{2} \)
43 \( 1 - 7.54T + 43T^{2} \)
47 \( 1 + 11.3T + 47T^{2} \)
53 \( 1 + 10.8T + 53T^{2} \)
59 \( 1 - 3.88T + 59T^{2} \)
61 \( 1 + 4.01T + 61T^{2} \)
67 \( 1 + 6.53T + 67T^{2} \)
71 \( 1 + 0.411T + 71T^{2} \)
73 \( 1 + 13.9T + 73T^{2} \)
79 \( 1 - 11.1T + 79T^{2} \)
83 \( 1 - 6.15T + 83T^{2} \)
89 \( 1 + 4.04T + 89T^{2} \)
97 \( 1 - 10.3T + 97T^{2} \)
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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

−7.61871782409178494809467470684, −6.55661156504915414727657478300, −6.18467423373851886055354599101, −5.47067709502495375422485714782, −4.65415310107979136225911915983, −4.07361393800499387324790832171, −3.12659056681963479877969488030, −2.08031465828343288149424535739, −1.27063393933418541834367086869, 0, 1.27063393933418541834367086869, 2.08031465828343288149424535739, 3.12659056681963479877969488030, 4.07361393800499387324790832171, 4.65415310107979136225911915983, 5.47067709502495375422485714782, 6.18467423373851886055354599101, 6.55661156504915414727657478300, 7.61871782409178494809467470684

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