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  − 1.42·3-s − 3.43·5-s + 0.534·7-s − 0.981·9-s + 2.02·11-s − 1.33·13-s + 4.87·15-s − 2.74·17-s − 2.29·19-s − 0.759·21-s − 7.13·23-s + 6.77·25-s + 5.65·27-s − 7.32·29-s + 1.25·31-s − 2.87·33-s − 1.83·35-s − 2.73·37-s + 1.89·39-s − 10.4·41-s + 6.16·43-s + 3.36·45-s + 5.32·47-s − 6.71·49-s + 3.90·51-s + 2.73·53-s − 6.93·55-s + ⋯
L(s)  = 1  − 0.820·3-s − 1.53·5-s + 0.202·7-s − 0.327·9-s + 0.609·11-s − 0.369·13-s + 1.25·15-s − 0.666·17-s − 0.526·19-s − 0.165·21-s − 1.48·23-s + 1.35·25-s + 1.08·27-s − 1.36·29-s + 0.225·31-s − 0.499·33-s − 0.309·35-s − 0.450·37-s + 0.303·39-s − 1.63·41-s + 0.940·43-s + 0.501·45-s + 0.776·47-s − 0.959·49-s + 0.546·51-s + 0.375·53-s − 0.934·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.2480091112$
$L(\frac12)$  $\approx$  $0.2480091112$
$L(\frac{3}{2})$   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 \( 1 + 1.42T + 3T^{2} \)
5 \( 1 + 3.43T + 5T^{2} \)
7 \( 1 - 0.534T + 7T^{2} \)
11 \( 1 - 2.02T + 11T^{2} \)
13 \( 1 + 1.33T + 13T^{2} \)
17 \( 1 + 2.74T + 17T^{2} \)
19 \( 1 + 2.29T + 19T^{2} \)
23 \( 1 + 7.13T + 23T^{2} \)
29 \( 1 + 7.32T + 29T^{2} \)
31 \( 1 - 1.25T + 31T^{2} \)
37 \( 1 + 2.73T + 37T^{2} \)
41 \( 1 + 10.4T + 41T^{2} \)
43 \( 1 - 6.16T + 43T^{2} \)
47 \( 1 - 5.32T + 47T^{2} \)
53 \( 1 - 2.73T + 53T^{2} \)
59 \( 1 - 2.37T + 59T^{2} \)
61 \( 1 + 2.88T + 61T^{2} \)
67 \( 1 + 7.96T + 67T^{2} \)
71 \( 1 + 9.75T + 71T^{2} \)
73 \( 1 - 2.03T + 73T^{2} \)
79 \( 1 - 2.43T + 79T^{2} \)
83 \( 1 + 16.3T + 83T^{2} \)
89 \( 1 + 1.73T + 89T^{2} \)
97 \( 1 + 10.8T + 97T^{2} \)
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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

−8.088909588323470449944451109553, −7.32944307225093356774495015781, −6.71117430930890663869246599431, −5.95524787520731945056306984889, −5.19623727069417837843793450575, −4.28419882489754525086114402767, −3.94917480190156745449869861850, −2.90936470382838248352720513349, −1.71016717438883994137427125651, −0.26665851485219583932389785447, 0.26665851485219583932389785447, 1.71016717438883994137427125651, 2.90936470382838248352720513349, 3.94917480190156745449869861850, 4.28419882489754525086114402767, 5.19623727069417837843793450575, 5.95524787520731945056306984889, 6.71117430930890663869246599431, 7.32944307225093356774495015781, 8.088909588323470449944451109553

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