Properties

Degree 2
Conductor $ 2^{2} \cdot 3 \cdot 23 \cdot 29 $
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 − 2.36·5-s + 2.82·7-s + 9-s + 2.46·11-s − 2.24·13-s + 2.36·15-s − 1.17·17-s + 5.00·19-s − 2.82·21-s − 23-s + 0.575·25-s − 27-s + 29-s − 6.16·31-s − 2.46·33-s − 6.66·35-s + 4.43·37-s + 2.24·39-s − 3.21·41-s − 12.1·43-s − 2.36·45-s − 3.40·47-s + 0.961·49-s + 1.17·51-s − 3.20·53-s − 5.81·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.05·5-s + 1.06·7-s + 0.333·9-s + 0.742·11-s − 0.622·13-s + 0.609·15-s − 0.285·17-s + 1.14·19-s − 0.615·21-s − 0.208·23-s + 0.115·25-s − 0.192·27-s + 0.185·29-s − 1.10·31-s − 0.428·33-s − 1.12·35-s + 0.728·37-s + 0.359·39-s − 0.502·41-s − 1.84·43-s − 0.351·45-s − 0.497·47-s + 0.137·49-s + 0.164·51-s − 0.440·53-s − 0.784·55-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

\( d \)  =  \(2\)
\( N \)  =  \(8004\)    =    \(2^{2} \cdot 3 \cdot 23 \cdot 29\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{8004} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  \(1\)
Selberg data  =  \((2,\ 8004,\ (\ :1/2),\ -1)\)
\(L(1)\)  \(=\)  \(0\)
\(L(\frac12)\)  \(=\)  \(0\)
\(L(\frac{3}{2})\)   not available
\(L(1)\)   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;3,\;23,\;29\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;3,\;23,\;29\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 - T \)
good5 \( 1 + 2.36T + 5T^{2} \)
7 \( 1 - 2.82T + 7T^{2} \)
11 \( 1 - 2.46T + 11T^{2} \)
13 \( 1 + 2.24T + 13T^{2} \)
17 \( 1 + 1.17T + 17T^{2} \)
19 \( 1 - 5.00T + 19T^{2} \)
31 \( 1 + 6.16T + 31T^{2} \)
37 \( 1 - 4.43T + 37T^{2} \)
41 \( 1 + 3.21T + 41T^{2} \)
43 \( 1 + 12.1T + 43T^{2} \)
47 \( 1 + 3.40T + 47T^{2} \)
53 \( 1 + 3.20T + 53T^{2} \)
59 \( 1 + 2.13T + 59T^{2} \)
61 \( 1 - 5.45T + 61T^{2} \)
67 \( 1 - 11.5T + 67T^{2} \)
71 \( 1 - 6.80T + 71T^{2} \)
73 \( 1 - 0.535T + 73T^{2} \)
79 \( 1 + 9.08T + 79T^{2} \)
83 \( 1 + 13.4T + 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 - 18.3T + 97T^{2} \)
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\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.46838499845374855675632438077, −6.95487710959489189953455499495, −6.12469414713321019825033200042, −5.13965885600448060129157785997, −4.81673904679158849909831817437, −3.95554249193733834980970151031, −3.33949274848023476873671345635, −2.04762520381333540327796156913, −1.17371600385289728730374795885, 0, 1.17371600385289728730374795885, 2.04762520381333540327796156913, 3.33949274848023476873671345635, 3.95554249193733834980970151031, 4.81673904679158849909831817437, 5.13965885600448060129157785997, 6.12469414713321019825033200042, 6.95487710959489189953455499495, 7.46838499845374855675632438077

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