Properties

Degree 2
Conductor $ 2^{3} \cdot 751 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.0243·3-s + 3.64·5-s − 4.79·7-s − 2.99·9-s + 4.87·11-s + 2.44·13-s − 0.0887·15-s − 2.69·17-s − 1.00·19-s + 0.116·21-s − 2.85·23-s + 8.29·25-s + 0.145·27-s + 5.94·29-s + 3.27·31-s − 0.118·33-s − 17.4·35-s + 6.54·37-s − 0.0594·39-s − 8.68·41-s + 2.95·43-s − 10.9·45-s − 10.3·47-s + 16.0·49-s + 0.0654·51-s + 7.10·53-s + 17.7·55-s + ⋯
L(s)  = 1  − 0.0140·3-s + 1.63·5-s − 1.81·7-s − 0.999·9-s + 1.46·11-s + 0.677·13-s − 0.0229·15-s − 0.652·17-s − 0.229·19-s + 0.0254·21-s − 0.595·23-s + 1.65·25-s + 0.0280·27-s + 1.10·29-s + 0.588·31-s − 0.0206·33-s − 2.95·35-s + 1.07·37-s − 0.00952·39-s − 1.35·41-s + 0.450·43-s − 1.63·45-s − 1.50·47-s + 2.28·49-s + 0.00917·51-s + 0.976·53-s + 2.39·55-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  =  0
Selberg data  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.156179076$
$L(\frac12)$  $\approx$  $2.156179076$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
751 \( 1 + T \)
good3 \( 1 + 0.0243T + 3T^{2} \)
5 \( 1 - 3.64T + 5T^{2} \)
7 \( 1 + 4.79T + 7T^{2} \)
11 \( 1 - 4.87T + 11T^{2} \)
13 \( 1 - 2.44T + 13T^{2} \)
17 \( 1 + 2.69T + 17T^{2} \)
19 \( 1 + 1.00T + 19T^{2} \)
23 \( 1 + 2.85T + 23T^{2} \)
29 \( 1 - 5.94T + 29T^{2} \)
31 \( 1 - 3.27T + 31T^{2} \)
37 \( 1 - 6.54T + 37T^{2} \)
41 \( 1 + 8.68T + 41T^{2} \)
43 \( 1 - 2.95T + 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 - 7.10T + 53T^{2} \)
59 \( 1 - 8.34T + 59T^{2} \)
61 \( 1 - 8.25T + 61T^{2} \)
67 \( 1 - 3.82T + 67T^{2} \)
71 \( 1 + 12.3T + 71T^{2} \)
73 \( 1 + 12.9T + 73T^{2} \)
79 \( 1 + 7.35T + 79T^{2} \)
83 \( 1 - 4.97T + 83T^{2} \)
89 \( 1 - 10.6T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−8.438440020048345451147618102251, −6.93947216220679991934461355990, −6.35560414682283571369796168217, −6.23111749775448469110209065258, −5.54580806226269267806712452989, −4.39075006495639563827417518953, −3.45842897425522005448933099313, −2.79522019973996374815947022048, −1.94872983586619557308696313500, −0.76490214853290545362251909385, 0.76490214853290545362251909385, 1.94872983586619557308696313500, 2.79522019973996374815947022048, 3.45842897425522005448933099313, 4.39075006495639563827417518953, 5.54580806226269267806712452989, 6.23111749775448469110209065258, 6.35560414682283571369796168217, 6.93947216220679991934461355990, 8.438440020048345451147618102251

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