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  − 0.684·3-s + 2.14·5-s + 4.94·7-s − 2.53·9-s + 5.83·11-s − 0.179·13-s − 1.46·15-s + 3.12·17-s + 1.14·19-s − 3.39·21-s − 6.64·23-s − 0.396·25-s + 3.78·27-s + 0.232·29-s + 7.29·31-s − 4.00·33-s + 10.6·35-s − 5.55·37-s + 0.123·39-s + 1.51·41-s + 2.08·43-s − 5.43·45-s + 9.61·47-s + 17.4·49-s − 2.13·51-s + 10.2·53-s + 12.5·55-s + ⋯
L(s)  = 1  − 0.395·3-s + 0.959·5-s + 1.87·7-s − 0.843·9-s + 1.76·11-s − 0.0499·13-s − 0.379·15-s + 0.756·17-s + 0.262·19-s − 0.739·21-s − 1.38·23-s − 0.0792·25-s + 0.729·27-s + 0.0432·29-s + 1.31·31-s − 0.696·33-s + 1.79·35-s − 0.913·37-s + 0.0197·39-s + 0.235·41-s + 0.317·43-s − 0.809·45-s + 1.40·47-s + 2.49·49-s − 0.299·51-s + 1.41·53-s + 1.68·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $3.024344861$
$L(\frac12)$  $\approx$  $3.024344861$
$L(\frac{3}{2})$   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 \( 1 + 0.684T + 3T^{2} \)
5 \( 1 - 2.14T + 5T^{2} \)
7 \( 1 - 4.94T + 7T^{2} \)
11 \( 1 - 5.83T + 11T^{2} \)
13 \( 1 + 0.179T + 13T^{2} \)
17 \( 1 - 3.12T + 17T^{2} \)
19 \( 1 - 1.14T + 19T^{2} \)
23 \( 1 + 6.64T + 23T^{2} \)
29 \( 1 - 0.232T + 29T^{2} \)
31 \( 1 - 7.29T + 31T^{2} \)
37 \( 1 + 5.55T + 37T^{2} \)
41 \( 1 - 1.51T + 41T^{2} \)
43 \( 1 - 2.08T + 43T^{2} \)
47 \( 1 - 9.61T + 47T^{2} \)
53 \( 1 - 10.2T + 53T^{2} \)
59 \( 1 + 7.92T + 59T^{2} \)
61 \( 1 + 0.773T + 61T^{2} \)
67 \( 1 + 7.77T + 67T^{2} \)
71 \( 1 - 1.12T + 71T^{2} \)
73 \( 1 + 10.6T + 73T^{2} \)
79 \( 1 - 3.02T + 79T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 + 12.5T + 89T^{2} \)
97 \( 1 - 6.78T + 97T^{2} \)
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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

−8.117512785695210050536199847369, −7.44652394221682599918314497007, −6.45066980219569828603040050130, −5.84910082745992182581745259807, −5.38427142929293296597100537663, −4.51331497115908526987626060133, −3.83844399013387904367112167889, −2.54874547485091919914442033681, −1.69691035619457213781607898290, −1.04353351550963445857462581824, 1.04353351550963445857462581824, 1.69691035619457213781607898290, 2.54874547485091919914442033681, 3.83844399013387904367112167889, 4.51331497115908526987626060133, 5.38427142929293296597100537663, 5.84910082745992182581745259807, 6.45066980219569828603040050130, 7.44652394221682599918314497007, 8.117512785695210050536199847369

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