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  − 2.49·3-s − 3.92·5-s + 2.18·7-s + 3.21·9-s + 3.47·11-s + 3.85·13-s + 9.78·15-s − 2.14·17-s + 7.07·19-s − 5.44·21-s − 0.572·23-s + 10.4·25-s − 0.530·27-s + 5.29·29-s + 8.09·31-s − 8.66·33-s − 8.58·35-s + 4.15·37-s − 9.61·39-s + 0.245·41-s + 5.43·43-s − 12.6·45-s + 1.50·47-s − 2.22·49-s + 5.35·51-s − 7.38·53-s − 13.6·55-s + ⋯
L(s)  = 1  − 1.43·3-s − 1.75·5-s + 0.826·7-s + 1.07·9-s + 1.04·11-s + 1.06·13-s + 2.52·15-s − 0.520·17-s + 1.62·19-s − 1.18·21-s − 0.119·23-s + 2.08·25-s − 0.102·27-s + 0.982·29-s + 1.45·31-s − 1.50·33-s − 1.45·35-s + 0.683·37-s − 1.53·39-s + 0.0382·41-s + 0.828·43-s − 1.88·45-s + 0.218·47-s − 0.317·49-s + 0.749·51-s − 1.01·53-s − 1.84·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.124318679$
$L(\frac12)$  $\approx$  $1.124318679$
$L(\frac{3}{2})$   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 \( 1 + 2.49T + 3T^{2} \)
5 \( 1 + 3.92T + 5T^{2} \)
7 \( 1 - 2.18T + 7T^{2} \)
11 \( 1 - 3.47T + 11T^{2} \)
13 \( 1 - 3.85T + 13T^{2} \)
17 \( 1 + 2.14T + 17T^{2} \)
19 \( 1 - 7.07T + 19T^{2} \)
23 \( 1 + 0.572T + 23T^{2} \)
29 \( 1 - 5.29T + 29T^{2} \)
31 \( 1 - 8.09T + 31T^{2} \)
37 \( 1 - 4.15T + 37T^{2} \)
41 \( 1 - 0.245T + 41T^{2} \)
43 \( 1 - 5.43T + 43T^{2} \)
47 \( 1 - 1.50T + 47T^{2} \)
53 \( 1 + 7.38T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 - 8.32T + 61T^{2} \)
67 \( 1 - 5.93T + 67T^{2} \)
71 \( 1 + 6.26T + 71T^{2} \)
73 \( 1 - 0.909T + 73T^{2} \)
79 \( 1 + 9.07T + 79T^{2} \)
83 \( 1 - 12.2T + 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 + 7.29T + 97T^{2} \)
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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

−7.996645708832103019326712625195, −7.35771902626778793173709519537, −6.56007737673676964657952142090, −6.08306511768089508634513864709, −5.01132477067290910787592032199, −4.54388180519976692622655625129, −3.91302622248643017202298359618, −3.03876658572444289471213391055, −1.25203912436925629832124333926, −0.73225474071909929962841319570, 0.73225474071909929962841319570, 1.25203912436925629832124333926, 3.03876658572444289471213391055, 3.91302622248643017202298359618, 4.54388180519976692622655625129, 5.01132477067290910787592032199, 6.08306511768089508634513864709, 6.56007737673676964657952142090, 7.35771902626778793173709519537, 7.996645708832103019326712625195

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