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 − 2.03·5-s − 0.0739·7-s − 2.53·9-s − 6.59·11-s − 2.19·13-s + 1.38·15-s − 2.82·17-s + 3.57·19-s + 0.0505·21-s − 3.59·23-s − 0.870·25-s + 3.78·27-s − 0.888·29-s − 3.39·31-s + 4.51·33-s + 0.150·35-s − 1.90·37-s + 1.49·39-s − 5.01·41-s − 8.72·43-s + 5.14·45-s − 7.66·47-s − 6.99·49-s + 1.93·51-s − 7.04·53-s + 13.4·55-s + ⋯
L(s)  = 1  − 0.394·3-s − 0.908·5-s − 0.0279·7-s − 0.844·9-s − 1.98·11-s − 0.607·13-s + 0.358·15-s − 0.684·17-s + 0.820·19-s + 0.0110·21-s − 0.750·23-s − 0.174·25-s + 0.728·27-s − 0.164·29-s − 0.609·31-s + 0.785·33-s + 0.0253·35-s − 0.312·37-s + 0.240·39-s − 0.783·41-s − 1.33·43-s + 0.767·45-s − 1.11·47-s − 0.999·49-s + 0.270·51-s − 0.967·53-s + 1.80·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.04873569368$
$L(\frac12)$  $\approx$  $0.04873569368$
$L(\frac{3}{2})$   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 \( 1 + 2.03T + 5T^{2} \)
7 \( 1 + 0.0739T + 7T^{2} \)
11 \( 1 + 6.59T + 11T^{2} \)
13 \( 1 + 2.19T + 13T^{2} \)
17 \( 1 + 2.82T + 17T^{2} \)
19 \( 1 - 3.57T + 19T^{2} \)
23 \( 1 + 3.59T + 23T^{2} \)
29 \( 1 + 0.888T + 29T^{2} \)
31 \( 1 + 3.39T + 31T^{2} \)
37 \( 1 + 1.90T + 37T^{2} \)
41 \( 1 + 5.01T + 41T^{2} \)
43 \( 1 + 8.72T + 43T^{2} \)
47 \( 1 + 7.66T + 47T^{2} \)
53 \( 1 + 7.04T + 53T^{2} \)
59 \( 1 + 1.37T + 59T^{2} \)
61 \( 1 - 0.987T + 61T^{2} \)
67 \( 1 + 0.134T + 67T^{2} \)
71 \( 1 - 3.40T + 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 + 13.9T + 79T^{2} \)
83 \( 1 + 0.111T + 83T^{2} \)
89 \( 1 + 0.114T + 89T^{2} \)
97 \( 1 + 14.8T + 97T^{2} \)
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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

−8.147831298594353043931622936540, −7.47812424236533898059674913943, −6.74738573767432012509678226059, −5.79892903387767055635445160195, −5.14569922198280687998028327111, −4.68916129841040955523777265465, −3.50072506582552975749759898382, −2.90446084141600226655154658551, −1.94848559041024908230089954388, −0.10934809479731050087755952967, 0.10934809479731050087755952967, 1.94848559041024908230089954388, 2.90446084141600226655154658551, 3.50072506582552975749759898382, 4.68916129841040955523777265465, 5.14569922198280687998028327111, 5.79892903387767055635445160195, 6.74738573767432012509678226059, 7.47812424236533898059674913943, 8.147831298594353043931622936540

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