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 − 2.52·5-s − 2.76·7-s + 9-s + 0.526·11-s − 1.93·13-s + 2.52·15-s + 0.739·17-s − 1.99·19-s + 2.76·21-s + 23-s + 1.37·25-s − 27-s − 29-s + 3.82·31-s − 0.526·33-s + 6.98·35-s + 0.747·37-s + 1.93·39-s + 3.52·41-s + 2.13·43-s − 2.52·45-s + 7.82·47-s + 0.664·49-s − 0.739·51-s + 7.39·53-s − 1.32·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.12·5-s − 1.04·7-s + 0.333·9-s + 0.158·11-s − 0.535·13-s + 0.651·15-s + 0.179·17-s − 0.458·19-s + 0.604·21-s + 0.208·23-s + 0.274·25-s − 0.192·27-s − 0.185·29-s + 0.686·31-s − 0.0916·33-s + 1.18·35-s + 0.122·37-s + 0.309·39-s + 0.551·41-s + 0.326·43-s − 0.376·45-s + 1.14·47-s + 0.0949·49-s − 0.103·51-s + 1.01·53-s − 0.179·55-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 \( 1 + 2.52T + 5T^{2} \)
7 \( 1 + 2.76T + 7T^{2} \)
11 \( 1 - 0.526T + 11T^{2} \)
13 \( 1 + 1.93T + 13T^{2} \)
17 \( 1 - 0.739T + 17T^{2} \)
19 \( 1 + 1.99T + 19T^{2} \)
31 \( 1 - 3.82T + 31T^{2} \)
37 \( 1 - 0.747T + 37T^{2} \)
41 \( 1 - 3.52T + 41T^{2} \)
43 \( 1 - 2.13T + 43T^{2} \)
47 \( 1 - 7.82T + 47T^{2} \)
53 \( 1 - 7.39T + 53T^{2} \)
59 \( 1 - 12.9T + 59T^{2} \)
61 \( 1 - 10.0T + 61T^{2} \)
67 \( 1 + 3.29T + 67T^{2} \)
71 \( 1 + 14.3T + 71T^{2} \)
73 \( 1 + 4.34T + 73T^{2} \)
79 \( 1 + 0.963T + 79T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 + 1.29T + 89T^{2} \)
97 \( 1 - 3.48T + 97T^{2} \)
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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

−7.19536234341841501741189017556, −7.03958929670737342591072730525, −6.07312501628020345153824743251, −5.52337119430782352595667130962, −4.46862450888024038610549439665, −4.03080459460370359481192229062, −3.21123490560750737882430352317, −2.35695805378065866472814312627, −0.895910024536567704333192372342, 0, 0.895910024536567704333192372342, 2.35695805378065866472814312627, 3.21123490560750737882430352317, 4.03080459460370359481192229062, 4.46862450888024038610549439665, 5.52337119430782352595667130962, 6.07312501628020345153824743251, 7.03958929670737342591072730525, 7.19536234341841501741189017556

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