Properties

Degree $2$
Conductor $8004$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2.58·5-s − 3.26·7-s + 9-s − 2.90·11-s + 5.43·13-s + 2.58·15-s − 2.76·17-s − 6.15·19-s + 3.26·21-s + 23-s + 1.68·25-s − 27-s + 29-s − 1.72·31-s + 2.90·33-s + 8.44·35-s − 4.97·37-s − 5.43·39-s − 6.02·41-s − 10.9·43-s − 2.58·45-s − 3.19·47-s + 3.67·49-s + 2.76·51-s + 5.41·53-s + 7.49·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.15·5-s − 1.23·7-s + 0.333·9-s − 0.874·11-s + 1.50·13-s + 0.667·15-s − 0.671·17-s − 1.41·19-s + 0.712·21-s + 0.208·23-s + 0.336·25-s − 0.192·27-s + 0.185·29-s − 0.310·31-s + 0.505·33-s + 1.42·35-s − 0.817·37-s − 0.870·39-s − 0.940·41-s − 1.67·43-s − 0.385·45-s − 0.466·47-s + 0.524·49-s + 0.387·51-s + 0.743·53-s + 1.01·55-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: \(0\)
Selberg data: \((2,\ 8004,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1638149716\)
\(L(\frac12)\) \(\approx\) \(0.1638149716\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.58T + 5T^{2} \)
7 \( 1 + 3.26T + 7T^{2} \)
11 \( 1 + 2.90T + 11T^{2} \)
13 \( 1 - 5.43T + 13T^{2} \)
17 \( 1 + 2.76T + 17T^{2} \)
19 \( 1 + 6.15T + 19T^{2} \)
31 \( 1 + 1.72T + 31T^{2} \)
37 \( 1 + 4.97T + 37T^{2} \)
41 \( 1 + 6.02T + 41T^{2} \)
43 \( 1 + 10.9T + 43T^{2} \)
47 \( 1 + 3.19T + 47T^{2} \)
53 \( 1 - 5.41T + 53T^{2} \)
59 \( 1 + 3.48T + 59T^{2} \)
61 \( 1 + 3.77T + 61T^{2} \)
67 \( 1 + 6.87T + 67T^{2} \)
71 \( 1 - 3.47T + 71T^{2} \)
73 \( 1 + 5.99T + 73T^{2} \)
79 \( 1 - 0.609T + 79T^{2} \)
83 \( 1 + 6.42T + 83T^{2} \)
89 \( 1 + 15.0T + 89T^{2} \)
97 \( 1 + 6.60T + 97T^{2} \)
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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

−7.85535983887893523852919477336, −6.93444461811398940396544854718, −6.54199936530244721243770351572, −5.89520193984548712454779427303, −5.00479618075985321941839262356, −4.17004710432571091631172306770, −3.60890783856492801421708426055, −2.90574579363686443297279496124, −1.64969236014087390588551429415, −0.20151237865175211205558153971, 0.20151237865175211205558153971, 1.64969236014087390588551429415, 2.90574579363686443297279496124, 3.60890783856492801421708426055, 4.17004710432571091631172306770, 5.00479618075985321941839262356, 5.89520193984548712454779427303, 6.54199936530244721243770351572, 6.93444461811398940396544854718, 7.85535983887893523852919477336

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