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  − 2.97·3-s + 0.343·5-s − 1.43·7-s + 5.82·9-s − 4.62·11-s − 0.0491·13-s − 1.02·15-s − 3.66·17-s − 8.44·19-s + 4.26·21-s − 7.46·23-s − 4.88·25-s − 8.40·27-s + 2.21·29-s + 0.764·31-s + 13.7·33-s − 0.493·35-s + 1.43·37-s + 0.146·39-s − 4.38·41-s + 0.508·43-s + 2.00·45-s + 1.72·47-s − 4.93·49-s + 10.8·51-s − 2.75·53-s − 1.58·55-s + ⋯
L(s)  = 1  − 1.71·3-s + 0.153·5-s − 0.542·7-s + 1.94·9-s − 1.39·11-s − 0.0136·13-s − 0.263·15-s − 0.889·17-s − 1.93·19-s + 0.931·21-s − 1.55·23-s − 0.976·25-s − 1.61·27-s + 0.410·29-s + 0.137·31-s + 2.39·33-s − 0.0834·35-s + 0.236·37-s + 0.0233·39-s − 0.685·41-s + 0.0775·43-s + 0.298·45-s + 0.252·47-s − 0.705·49-s + 1.52·51-s − 0.378·53-s − 0.214·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.02576953992$
$L(\frac12)$  $\approx$  $0.02576953992$
$L(\frac{3}{2})$   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 \( 1 + 2.97T + 3T^{2} \)
5 \( 1 - 0.343T + 5T^{2} \)
7 \( 1 + 1.43T + 7T^{2} \)
11 \( 1 + 4.62T + 11T^{2} \)
13 \( 1 + 0.0491T + 13T^{2} \)
17 \( 1 + 3.66T + 17T^{2} \)
19 \( 1 + 8.44T + 19T^{2} \)
23 \( 1 + 7.46T + 23T^{2} \)
29 \( 1 - 2.21T + 29T^{2} \)
31 \( 1 - 0.764T + 31T^{2} \)
37 \( 1 - 1.43T + 37T^{2} \)
41 \( 1 + 4.38T + 41T^{2} \)
43 \( 1 - 0.508T + 43T^{2} \)
47 \( 1 - 1.72T + 47T^{2} \)
53 \( 1 + 2.75T + 53T^{2} \)
59 \( 1 + 3.37T + 59T^{2} \)
61 \( 1 - 4.47T + 61T^{2} \)
67 \( 1 - 8.03T + 67T^{2} \)
71 \( 1 + 7.02T + 71T^{2} \)
73 \( 1 + 12.2T + 73T^{2} \)
79 \( 1 + 16.4T + 79T^{2} \)
83 \( 1 + 2.64T + 83T^{2} \)
89 \( 1 - 4.57T + 89T^{2} \)
97 \( 1 - 0.346T + 97T^{2} \)
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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

−8.006647737959040456923569482584, −7.15077875687901014704971341231, −6.33920375917943808570541329899, −6.09151813092900133671098324052, −5.33943123591650000237954878233, −4.54451764540599016731761489514, −4.01049194228207662342121313160, −2.60506905997615599692185752912, −1.76197764539626279226953094585, −0.087436300995142246084970559986, 0.087436300995142246084970559986, 1.76197764539626279226953094585, 2.60506905997615599692185752912, 4.01049194228207662342121313160, 4.54451764540599016731761489514, 5.33943123591650000237954878233, 6.09151813092900133671098324052, 6.33920375917943808570541329899, 7.15077875687901014704971341231, 8.006647737959040456923569482584

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