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 − 3.70·5-s − 1.59·7-s + 9-s − 2.15·11-s − 3.62·13-s + 3.70·15-s + 6.51·17-s − 5.86·19-s + 1.59·21-s − 23-s + 8.74·25-s − 27-s + 29-s + 10.0·31-s + 2.15·33-s + 5.91·35-s + 1.99·37-s + 3.62·39-s − 2.83·41-s − 1.53·43-s − 3.70·45-s − 1.23·47-s − 4.45·49-s − 6.51·51-s − 7.71·53-s + 7.97·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.65·5-s − 0.602·7-s + 0.333·9-s − 0.648·11-s − 1.00·13-s + 0.957·15-s + 1.58·17-s − 1.34·19-s + 0.347·21-s − 0.208·23-s + 1.74·25-s − 0.192·27-s + 0.185·29-s + 1.80·31-s + 0.374·33-s + 0.999·35-s + 0.327·37-s + 0.581·39-s − 0.443·41-s − 0.233·43-s − 0.552·45-s − 0.180·47-s − 0.636·49-s − 0.912·51-s − 1.05·53-s + 1.07·55-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 \( 1 + 3.70T + 5T^{2} \)
7 \( 1 + 1.59T + 7T^{2} \)
11 \( 1 + 2.15T + 11T^{2} \)
13 \( 1 + 3.62T + 13T^{2} \)
17 \( 1 - 6.51T + 17T^{2} \)
19 \( 1 + 5.86T + 19T^{2} \)
31 \( 1 - 10.0T + 31T^{2} \)
37 \( 1 - 1.99T + 37T^{2} \)
41 \( 1 + 2.83T + 41T^{2} \)
43 \( 1 + 1.53T + 43T^{2} \)
47 \( 1 + 1.23T + 47T^{2} \)
53 \( 1 + 7.71T + 53T^{2} \)
59 \( 1 - 7.21T + 59T^{2} \)
61 \( 1 - 1.77T + 61T^{2} \)
67 \( 1 - 6.75T + 67T^{2} \)
71 \( 1 - 6.62T + 71T^{2} \)
73 \( 1 - 9.42T + 73T^{2} \)
79 \( 1 - 14.8T + 79T^{2} \)
83 \( 1 - 11.5T + 83T^{2} \)
89 \( 1 + 9.06T + 89T^{2} \)
97 \( 1 - 0.456T + 97T^{2} \)
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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

−7.67943196870374410314239120162, −6.73725455508478692220690071216, −6.31195310712468253177740198689, −5.17248031573011044090756458775, −4.74767361228672635931110269847, −3.89483256368359709158132855319, −3.25833081807587787369817275344, −2.38578916702211903478329458008, −0.828119212387202146088289562628, 0, 0.828119212387202146088289562628, 2.38578916702211903478329458008, 3.25833081807587787369817275344, 3.89483256368359709158132855319, 4.74767361228672635931110269847, 5.17248031573011044090756458775, 6.31195310712468253177740198689, 6.73725455508478692220690071216, 7.67943196870374410314239120162

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