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  + 1.28·3-s − 3.57·5-s − 2.67·7-s − 1.34·9-s − 2.04·11-s − 0.441·13-s − 4.60·15-s − 5.96·17-s − 6.28·19-s − 3.44·21-s − 5.22·23-s + 7.75·25-s − 5.59·27-s + 9.50·29-s + 0.189·31-s − 2.63·33-s + 9.55·35-s − 9.10·37-s − 0.568·39-s + 7.36·41-s − 2.12·43-s + 4.78·45-s + 0.117·47-s + 0.159·49-s − 7.68·51-s − 7.37·53-s + 7.29·55-s + ⋯
L(s)  = 1  + 0.743·3-s − 1.59·5-s − 1.01·7-s − 0.446·9-s − 0.615·11-s − 0.122·13-s − 1.18·15-s − 1.44·17-s − 1.44·19-s − 0.752·21-s − 1.09·23-s + 1.55·25-s − 1.07·27-s + 1.76·29-s + 0.0340·31-s − 0.457·33-s + 1.61·35-s − 1.49·37-s − 0.0909·39-s + 1.15·41-s − 0.324·43-s + 0.713·45-s + 0.0171·47-s + 0.0227·49-s − 1.07·51-s − 1.01·53-s + 0.983·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.2853107665$
$L(\frac12)$  $\approx$  $0.2853107665$
$L(\frac{3}{2})$   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 \( 1 - 1.28T + 3T^{2} \)
5 \( 1 + 3.57T + 5T^{2} \)
7 \( 1 + 2.67T + 7T^{2} \)
11 \( 1 + 2.04T + 11T^{2} \)
13 \( 1 + 0.441T + 13T^{2} \)
17 \( 1 + 5.96T + 17T^{2} \)
19 \( 1 + 6.28T + 19T^{2} \)
23 \( 1 + 5.22T + 23T^{2} \)
29 \( 1 - 9.50T + 29T^{2} \)
31 \( 1 - 0.189T + 31T^{2} \)
37 \( 1 + 9.10T + 37T^{2} \)
41 \( 1 - 7.36T + 41T^{2} \)
43 \( 1 + 2.12T + 43T^{2} \)
47 \( 1 - 0.117T + 47T^{2} \)
53 \( 1 + 7.37T + 53T^{2} \)
59 \( 1 + 6.95T + 59T^{2} \)
61 \( 1 - 3.87T + 61T^{2} \)
67 \( 1 - 13.8T + 67T^{2} \)
71 \( 1 + 8.22T + 71T^{2} \)
73 \( 1 + 1.09T + 73T^{2} \)
79 \( 1 - 13.2T + 79T^{2} \)
83 \( 1 - 5.72T + 83T^{2} \)
89 \( 1 + 10.3T + 89T^{2} \)
97 \( 1 - 0.540T + 97T^{2} \)
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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

−8.194462114237794295668285529211, −7.56491113045073970664487397364, −6.63941464303123843389946406004, −6.26958004776054962273418757562, −4.96482799873232795895719686234, −4.22298703990623541141075609960, −3.63412429315226152961502188084, −2.86480986066460812464642355583, −2.19531179735062428772493422961, −0.24693572420488352311102082588, 0.24693572420488352311102082588, 2.19531179735062428772493422961, 2.86480986066460812464642355583, 3.63412429315226152961502188084, 4.22298703990623541141075609960, 4.96482799873232795895719686234, 6.26958004776054962273418757562, 6.63941464303123843389946406004, 7.56491113045073970664487397364, 8.194462114237794295668285529211

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