Properties

Degree 2
Conductor $ 2^{3} \cdot 751 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.67·3-s + 3.24·5-s − 2.55·7-s − 0.201·9-s − 3.86·11-s + 4.77·13-s + 5.43·15-s − 7.34·17-s + 5.13·19-s − 4.26·21-s − 4.94·23-s + 5.55·25-s − 5.35·27-s − 0.579·29-s − 6.31·31-s − 6.46·33-s − 8.29·35-s − 7.52·37-s + 7.98·39-s − 7.67·41-s − 6.46·43-s − 0.654·45-s − 12.1·47-s − 0.488·49-s − 12.2·51-s − 2.15·53-s − 12.5·55-s + ⋯
L(s)  = 1  + 0.965·3-s + 1.45·5-s − 0.964·7-s − 0.0671·9-s − 1.16·11-s + 1.32·13-s + 1.40·15-s − 1.78·17-s + 1.17·19-s − 0.931·21-s − 1.03·23-s + 1.11·25-s − 1.03·27-s − 0.107·29-s − 1.13·31-s − 1.12·33-s − 1.40·35-s − 1.23·37-s + 1.27·39-s − 1.19·41-s − 0.985·43-s − 0.0975·45-s − 1.77·47-s − 0.0697·49-s − 1.72·51-s − 0.295·53-s − 1.69·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

\( d \)  =  \(2\)
\( N \)  =  \(6008\)    =    \(2^{3} \cdot 751\)
\( \varepsilon \)  =  $-1$
motivic weight  =  \(1\)
character  :  $\chi_{6008} (1, \cdot )$
primitive  :  yes
self-dual  :  yes
analytic rank  =  1
Selberg data  =  $(2,\ 6008,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   not available
$L(1)$   not available

Euler product

\[L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1} \]where, for $p \notin \{2,\;751\}$,\[F_p(T) = 1 - a_p T + p T^2 .\]If $p \in \{2,\;751\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 + T \)
good3 \( 1 - 1.67T + 3T^{2} \)
5 \( 1 - 3.24T + 5T^{2} \)
7 \( 1 + 2.55T + 7T^{2} \)
11 \( 1 + 3.86T + 11T^{2} \)
13 \( 1 - 4.77T + 13T^{2} \)
17 \( 1 + 7.34T + 17T^{2} \)
19 \( 1 - 5.13T + 19T^{2} \)
23 \( 1 + 4.94T + 23T^{2} \)
29 \( 1 + 0.579T + 29T^{2} \)
31 \( 1 + 6.31T + 31T^{2} \)
37 \( 1 + 7.52T + 37T^{2} \)
41 \( 1 + 7.67T + 41T^{2} \)
43 \( 1 + 6.46T + 43T^{2} \)
47 \( 1 + 12.1T + 47T^{2} \)
53 \( 1 + 2.15T + 53T^{2} \)
59 \( 1 - 5.57T + 59T^{2} \)
61 \( 1 - 2.81T + 61T^{2} \)
67 \( 1 - 15.2T + 67T^{2} \)
71 \( 1 - 12.4T + 71T^{2} \)
73 \( 1 + 1.40T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 - 1.68T + 83T^{2} \)
89 \( 1 - 15.0T + 89T^{2} \)
97 \( 1 - 5.00T + 97T^{2} \)
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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

−7.927680399067412524957233832881, −6.82651742075360160672481719522, −6.39497599208673382778602429250, −5.58735870231829191400906557036, −5.05487644151094955271350281712, −3.63460562561125218706782133690, −3.23552777108726981752651094220, −2.24292387725038410280487453652, −1.78843453694972439419177084505, 0, 1.78843453694972439419177084505, 2.24292387725038410280487453652, 3.23552777108726981752651094220, 3.63460562561125218706782133690, 5.05487644151094955271350281712, 5.58735870231829191400906557036, 6.39497599208673382778602429250, 6.82651742075360160672481719522, 7.927680399067412524957233832881

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