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  − 0.0368·3-s − 3.59·5-s − 0.547·7-s − 2.99·9-s + 1.30·11-s + 0.987·13-s + 0.132·15-s + 1.11·17-s + 5.02·19-s + 0.0201·21-s − 2.20·23-s + 7.89·25-s + 0.220·27-s + 0.0479·29-s + 3.20·31-s − 0.0481·33-s + 1.96·35-s − 6.60·37-s − 0.0363·39-s − 0.410·41-s − 7.81·43-s + 10.7·45-s + 9.58·47-s − 6.70·49-s − 0.0411·51-s − 1.16·53-s − 4.69·55-s + ⋯
L(s)  = 1  − 0.0212·3-s − 1.60·5-s − 0.206·7-s − 0.999·9-s + 0.394·11-s + 0.273·13-s + 0.0341·15-s + 0.270·17-s + 1.15·19-s + 0.00439·21-s − 0.458·23-s + 1.57·25-s + 0.0425·27-s + 0.00889·29-s + 0.574·31-s − 0.00838·33-s + 0.332·35-s − 1.08·37-s − 0.00582·39-s − 0.0641·41-s − 1.19·43-s + 1.60·45-s + 1.39·47-s − 0.957·49-s − 0.00575·51-s − 0.160·53-s − 0.633·55-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 \( 1 + 0.0368T + 3T^{2} \)
5 \( 1 + 3.59T + 5T^{2} \)
7 \( 1 + 0.547T + 7T^{2} \)
11 \( 1 - 1.30T + 11T^{2} \)
13 \( 1 - 0.987T + 13T^{2} \)
17 \( 1 - 1.11T + 17T^{2} \)
19 \( 1 - 5.02T + 19T^{2} \)
23 \( 1 + 2.20T + 23T^{2} \)
29 \( 1 - 0.0479T + 29T^{2} \)
31 \( 1 - 3.20T + 31T^{2} \)
37 \( 1 + 6.60T + 37T^{2} \)
41 \( 1 + 0.410T + 41T^{2} \)
43 \( 1 + 7.81T + 43T^{2} \)
47 \( 1 - 9.58T + 47T^{2} \)
53 \( 1 + 1.16T + 53T^{2} \)
59 \( 1 - 7.75T + 59T^{2} \)
61 \( 1 - 8.47T + 61T^{2} \)
67 \( 1 - 5.28T + 67T^{2} \)
71 \( 1 - 14.7T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 - 5.56T + 79T^{2} \)
83 \( 1 + 0.439T + 83T^{2} \)
89 \( 1 - 11.3T + 89T^{2} \)
97 \( 1 + 17.5T + 97T^{2} \)
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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

−7.85734553468001663657487071764, −7.07554289188685919598009374410, −6.44067667838350824194449277904, −5.47687402335574663797917368917, −4.85159332251260518158263087307, −3.70802571009670846433013398435, −3.54150568891861405584390360843, −2.53400696618117246828279466387, −1.05415338382024604219967608969, 0, 1.05415338382024604219967608969, 2.53400696618117246828279466387, 3.54150568891861405584390360843, 3.70802571009670846433013398435, 4.85159332251260518158263087307, 5.47687402335574663797917368917, 6.44067667838350824194449277904, 7.07554289188685919598009374410, 7.85734553468001663657487071764

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