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  − 0.962·3-s + 2.73·5-s − 4.42·7-s − 2.07·9-s − 2.59·11-s + 2.44·13-s − 2.62·15-s + 0.359·17-s + 4.11·19-s + 4.26·21-s + 0.0487·23-s + 2.45·25-s + 4.88·27-s + 1.62·29-s − 1.82·31-s + 2.49·33-s − 12.0·35-s + 7.04·37-s − 2.35·39-s − 2.33·41-s − 0.598·43-s − 5.66·45-s + 12.9·47-s + 12.5·49-s − 0.345·51-s + 3.71·53-s − 7.07·55-s + ⋯
L(s)  = 1  − 0.555·3-s + 1.22·5-s − 1.67·7-s − 0.691·9-s − 0.780·11-s + 0.677·13-s − 0.678·15-s + 0.0870·17-s + 0.944·19-s + 0.929·21-s + 0.0101·23-s + 0.491·25-s + 0.939·27-s + 0.302·29-s − 0.327·31-s + 0.434·33-s − 2.04·35-s + 1.15·37-s − 0.376·39-s − 0.364·41-s − 0.0913·43-s − 0.843·45-s + 1.89·47-s + 1.79·49-s − 0.0483·51-s + 0.510·53-s − 0.953·55-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 \( 1 + 0.962T + 3T^{2} \)
5 \( 1 - 2.73T + 5T^{2} \)
7 \( 1 + 4.42T + 7T^{2} \)
11 \( 1 + 2.59T + 11T^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
17 \( 1 - 0.359T + 17T^{2} \)
19 \( 1 - 4.11T + 19T^{2} \)
23 \( 1 - 0.0487T + 23T^{2} \)
29 \( 1 - 1.62T + 29T^{2} \)
31 \( 1 + 1.82T + 31T^{2} \)
37 \( 1 - 7.04T + 37T^{2} \)
41 \( 1 + 2.33T + 41T^{2} \)
43 \( 1 + 0.598T + 43T^{2} \)
47 \( 1 - 12.9T + 47T^{2} \)
53 \( 1 - 3.71T + 53T^{2} \)
59 \( 1 - 0.779T + 59T^{2} \)
61 \( 1 + 12.1T + 61T^{2} \)
67 \( 1 + 5.58T + 67T^{2} \)
71 \( 1 + 14.3T + 71T^{2} \)
73 \( 1 - 10.0T + 73T^{2} \)
79 \( 1 + 5.23T + 79T^{2} \)
83 \( 1 + 11.6T + 83T^{2} \)
89 \( 1 - 13.4T + 89T^{2} \)
97 \( 1 + 6.27T + 97T^{2} \)
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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

−7.60160686255466022124076306121, −6.78538182163646109617514121668, −6.07312414120916492824183547425, −5.79284317264107314301580692691, −5.19158291961321740470336378628, −3.98123133970416205239696050648, −2.95645840734894056504325166317, −2.59963359471257012910610964354, −1.18154424234106845908492046374, 0, 1.18154424234106845908492046374, 2.59963359471257012910610964354, 2.95645840734894056504325166317, 3.98123133970416205239696050648, 5.19158291961321740470336378628, 5.79284317264107314301580692691, 6.07312414120916492824183547425, 6.78538182163646109617514121668, 7.60160686255466022124076306121

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