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  − 1.38·3-s − 1.32·5-s + 2.39·7-s − 1.08·9-s + 3.31·11-s − 3.52·13-s + 1.82·15-s + 2.72·17-s + 6.86·19-s − 3.31·21-s + 6.61·23-s − 3.25·25-s + 5.65·27-s − 7.66·29-s − 1.56·31-s − 4.58·33-s − 3.16·35-s + 1.61·37-s + 4.88·39-s + 3.10·41-s − 2.57·43-s + 1.43·45-s − 2.66·47-s − 1.26·49-s − 3.77·51-s + 3.91·53-s − 4.37·55-s + ⋯
L(s)  = 1  − 0.799·3-s − 0.590·5-s + 0.905·7-s − 0.360·9-s + 0.999·11-s − 0.978·13-s + 0.472·15-s + 0.660·17-s + 1.57·19-s − 0.723·21-s + 1.37·23-s − 0.650·25-s + 1.08·27-s − 1.42·29-s − 0.281·31-s − 0.798·33-s − 0.535·35-s + 0.265·37-s + 0.782·39-s + 0.484·41-s − 0.393·43-s + 0.213·45-s − 0.388·47-s − 0.180·49-s − 0.528·51-s + 0.537·53-s − 0.590·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.374016969$
$L(\frac12)$  $\approx$  $1.374016969$
$L(\frac{3}{2})$   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 \( 1 + 1.38T + 3T^{2} \)
5 \( 1 + 1.32T + 5T^{2} \)
7 \( 1 - 2.39T + 7T^{2} \)
11 \( 1 - 3.31T + 11T^{2} \)
13 \( 1 + 3.52T + 13T^{2} \)
17 \( 1 - 2.72T + 17T^{2} \)
19 \( 1 - 6.86T + 19T^{2} \)
23 \( 1 - 6.61T + 23T^{2} \)
29 \( 1 + 7.66T + 29T^{2} \)
31 \( 1 + 1.56T + 31T^{2} \)
37 \( 1 - 1.61T + 37T^{2} \)
41 \( 1 - 3.10T + 41T^{2} \)
43 \( 1 + 2.57T + 43T^{2} \)
47 \( 1 + 2.66T + 47T^{2} \)
53 \( 1 - 3.91T + 53T^{2} \)
59 \( 1 - 1.25T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 - 9.01T + 67T^{2} \)
71 \( 1 - 6.20T + 71T^{2} \)
73 \( 1 + 8.76T + 73T^{2} \)
79 \( 1 + 10.5T + 79T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 + 13.0T + 89T^{2} \)
97 \( 1 - 3.05T + 97T^{2} \)
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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

−7.944678562783609617124261272694, −7.31993546404641460755628105745, −6.82969225944639613774487147841, −5.62127941884277790451864660975, −5.37684386148589262933688021397, −4.58021195118303924095595776028, −3.71884456181469286983297367501, −2.88247044376578693535567575078, −1.61198759660071970950316271781, −0.67444187286535653805058292090, 0.67444187286535653805058292090, 1.61198759660071970950316271781, 2.88247044376578693535567575078, 3.71884456181469286983297367501, 4.58021195118303924095595776028, 5.37684386148589262933688021397, 5.62127941884277790451864660975, 6.82969225944639613774487147841, 7.31993546404641460755628105745, 7.944678562783609617124261272694

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