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  − 2.58·3-s + 0.0890·5-s + 4.21·7-s + 3.69·9-s − 5.56·11-s + 5.65·13-s − 0.230·15-s + 4.66·17-s + 5.92·19-s − 10.9·21-s − 2.98·23-s − 4.99·25-s − 1.79·27-s + 6.14·29-s − 0.725·31-s + 14.3·33-s + 0.375·35-s + 11.8·37-s − 14.6·39-s − 3.43·41-s + 4.18·43-s + 0.328·45-s − 5.47·47-s + 10.7·49-s − 12.0·51-s + 2.96·53-s − 0.494·55-s + ⋯
L(s)  = 1  − 1.49·3-s + 0.0398·5-s + 1.59·7-s + 1.23·9-s − 1.67·11-s + 1.56·13-s − 0.0594·15-s + 1.13·17-s + 1.35·19-s − 2.38·21-s − 0.621·23-s − 0.998·25-s − 0.344·27-s + 1.14·29-s − 0.130·31-s + 2.50·33-s + 0.0634·35-s + 1.94·37-s − 2.34·39-s − 0.537·41-s + 0.637·43-s + 0.0489·45-s − 0.798·47-s + 1.54·49-s − 1.68·51-s + 0.407·53-s − 0.0667·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.544226828$
$L(\frac12)$  $\approx$  $1.544226828$
$L(\frac{3}{2})$   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 \( 1 + 2.58T + 3T^{2} \)
5 \( 1 - 0.0890T + 5T^{2} \)
7 \( 1 - 4.21T + 7T^{2} \)
11 \( 1 + 5.56T + 11T^{2} \)
13 \( 1 - 5.65T + 13T^{2} \)
17 \( 1 - 4.66T + 17T^{2} \)
19 \( 1 - 5.92T + 19T^{2} \)
23 \( 1 + 2.98T + 23T^{2} \)
29 \( 1 - 6.14T + 29T^{2} \)
31 \( 1 + 0.725T + 31T^{2} \)
37 \( 1 - 11.8T + 37T^{2} \)
41 \( 1 + 3.43T + 41T^{2} \)
43 \( 1 - 4.18T + 43T^{2} \)
47 \( 1 + 5.47T + 47T^{2} \)
53 \( 1 - 2.96T + 53T^{2} \)
59 \( 1 - 3.19T + 59T^{2} \)
61 \( 1 - 2.66T + 61T^{2} \)
67 \( 1 + 2.11T + 67T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 + 5.71T + 73T^{2} \)
79 \( 1 - 7.85T + 79T^{2} \)
83 \( 1 + 2.69T + 83T^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + 4.06T + 97T^{2} \)
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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

−7.935923323360912253722741074686, −7.57070050645407342061746480111, −6.43396954368426841095494676678, −5.63781379692264166273400628397, −5.42606496778085991596818693477, −4.74171933321310363861847629206, −3.88190378401756187477812115924, −2.70661288323044448042416051106, −1.47181794034642435404738158899, −0.78812086024047647917355885705, 0.78812086024047647917355885705, 1.47181794034642435404738158899, 2.70661288323044448042416051106, 3.88190378401756187477812115924, 4.74171933321310363861847629206, 5.42606496778085991596818693477, 5.63781379692264166273400628397, 6.43396954368426841095494676678, 7.57070050645407342061746480111, 7.935923323360912253722741074686

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