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 − 0.123·5-s + 3.12·7-s + 9-s − 3.34·11-s − 4.64·13-s + 0.123·15-s − 4.27·17-s − 2.36·19-s − 3.12·21-s + 23-s − 4.98·25-s − 27-s + 29-s − 4.48·31-s + 3.34·33-s − 0.386·35-s − 5.19·37-s + 4.64·39-s − 7.96·41-s + 3.81·43-s − 0.123·45-s + 2.73·47-s + 2.74·49-s + 4.27·51-s + 11.8·53-s + 0.414·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.0554·5-s + 1.17·7-s + 0.333·9-s − 1.00·11-s − 1.28·13-s + 0.0319·15-s − 1.03·17-s − 0.543·19-s − 0.681·21-s + 0.208·23-s − 0.996·25-s − 0.192·27-s + 0.185·29-s − 0.804·31-s + 0.582·33-s − 0.0653·35-s − 0.854·37-s + 0.743·39-s − 1.24·41-s + 0.581·43-s − 0.0184·45-s + 0.399·47-s + 0.391·49-s + 0.599·51-s + 1.63·53-s + 0.0558·55-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

\(L(1)\) \(\approx\) \(1.053745139\)
\(L(\frac12)\) \(\approx\) \(1.053745139\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.123T + 5T^{2} \)
7 \( 1 - 3.12T + 7T^{2} \)
11 \( 1 + 3.34T + 11T^{2} \)
13 \( 1 + 4.64T + 13T^{2} \)
17 \( 1 + 4.27T + 17T^{2} \)
19 \( 1 + 2.36T + 19T^{2} \)
31 \( 1 + 4.48T + 31T^{2} \)
37 \( 1 + 5.19T + 37T^{2} \)
41 \( 1 + 7.96T + 41T^{2} \)
43 \( 1 - 3.81T + 43T^{2} \)
47 \( 1 - 2.73T + 47T^{2} \)
53 \( 1 - 11.8T + 53T^{2} \)
59 \( 1 - 11.8T + 59T^{2} \)
61 \( 1 - 1.29T + 61T^{2} \)
67 \( 1 - 13.6T + 67T^{2} \)
71 \( 1 + 14.5T + 71T^{2} \)
73 \( 1 + 1.55T + 73T^{2} \)
79 \( 1 - 9.29T + 79T^{2} \)
83 \( 1 - 16.0T + 83T^{2} \)
89 \( 1 - 9.47T + 89T^{2} \)
97 \( 1 - 11.4T + 97T^{2} \)
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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

−7.68910832715383192862711374835, −7.26174072588840149247286923850, −6.50416839572482266618331859774, −5.50358043707598025134267073232, −5.08096460709208519731056590239, −4.52876694305998410992404537550, −3.66460379087629168895903799740, −2.30336225918194400854270158396, −1.99806300006127544376876788705, −0.49903298795551450653559208768, 0.49903298795551450653559208768, 1.99806300006127544376876788705, 2.30336225918194400854270158396, 3.66460379087629168895903799740, 4.52876694305998410992404537550, 5.08096460709208519731056590239, 5.50358043707598025134267073232, 6.50416839572482266618331859774, 7.26174072588840149247286923850, 7.68910832715383192862711374835

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