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  + 1.25·3-s + 4.39·5-s − 3.72·7-s − 1.42·9-s + 0.335·11-s − 6.63·13-s + 5.51·15-s + 5.58·17-s − 3.46·19-s − 4.68·21-s − 2.62·23-s + 14.2·25-s − 5.55·27-s + 4.43·29-s − 10.1·31-s + 0.420·33-s − 16.3·35-s − 4.76·37-s − 8.32·39-s − 5.06·41-s − 4.97·43-s − 6.25·45-s + 0.633·47-s + 6.91·49-s + 7.00·51-s + 6.19·53-s + 1.47·55-s + ⋯
L(s)  = 1  + 0.724·3-s + 1.96·5-s − 1.40·7-s − 0.474·9-s + 0.101·11-s − 1.84·13-s + 1.42·15-s + 1.35·17-s − 0.795·19-s − 1.02·21-s − 0.547·23-s + 2.85·25-s − 1.06·27-s + 0.822·29-s − 1.81·31-s + 0.0732·33-s − 2.76·35-s − 0.784·37-s − 1.33·39-s − 0.790·41-s − 0.758·43-s − 0.932·45-s + 0.0923·47-s + 0.987·49-s + 0.981·51-s + 0.850·53-s + 0.198·55-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 \( 1 - 1.25T + 3T^{2} \)
5 \( 1 - 4.39T + 5T^{2} \)
7 \( 1 + 3.72T + 7T^{2} \)
11 \( 1 - 0.335T + 11T^{2} \)
13 \( 1 + 6.63T + 13T^{2} \)
17 \( 1 - 5.58T + 17T^{2} \)
19 \( 1 + 3.46T + 19T^{2} \)
23 \( 1 + 2.62T + 23T^{2} \)
29 \( 1 - 4.43T + 29T^{2} \)
31 \( 1 + 10.1T + 31T^{2} \)
37 \( 1 + 4.76T + 37T^{2} \)
41 \( 1 + 5.06T + 41T^{2} \)
43 \( 1 + 4.97T + 43T^{2} \)
47 \( 1 - 0.633T + 47T^{2} \)
53 \( 1 - 6.19T + 53T^{2} \)
59 \( 1 - 3.08T + 59T^{2} \)
61 \( 1 - 7.40T + 61T^{2} \)
67 \( 1 + 11.5T + 67T^{2} \)
71 \( 1 + 0.566T + 71T^{2} \)
73 \( 1 + 12.8T + 73T^{2} \)
79 \( 1 - 16.1T + 79T^{2} \)
83 \( 1 - 12.8T + 83T^{2} \)
89 \( 1 + 7.99T + 89T^{2} \)
97 \( 1 + 14.3T + 97T^{2} \)
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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

−7.69197630204019510470532359479, −6.85395472912404673733737771479, −6.37530896090370155418658091004, −5.46254584888459692322173690707, −5.24200020110704925534059161768, −3.79024367061829253233815012590, −2.91762053138016499222715955183, −2.48184001872843176341057752491, −1.66987998702997623954864098587, 0, 1.66987998702997623954864098587, 2.48184001872843176341057752491, 2.91762053138016499222715955183, 3.79024367061829253233815012590, 5.24200020110704925534059161768, 5.46254584888459692322173690707, 6.37530896090370155418658091004, 6.85395472912404673733737771479, 7.69197630204019510470532359479

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