Properties

Degree 2
Conductor $ 2^{3} \cdot 751 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.65·3-s + 2.02·5-s − 1.52·7-s − 0.272·9-s − 3.27·11-s + 0.0809·13-s + 3.34·15-s + 5.20·17-s − 4.81·19-s − 2.51·21-s − 0.265·23-s − 0.897·25-s − 5.40·27-s − 7.75·29-s − 1.01·31-s − 5.41·33-s − 3.08·35-s + 8.64·37-s + 0.133·39-s + 6.42·41-s − 7.71·43-s − 0.551·45-s − 6.87·47-s − 4.68·49-s + 8.60·51-s − 4.57·53-s − 6.64·55-s + ⋯
L(s)  = 1  + 0.953·3-s + 0.905·5-s − 0.574·7-s − 0.0907·9-s − 0.988·11-s + 0.0224·13-s + 0.863·15-s + 1.26·17-s − 1.10·19-s − 0.548·21-s − 0.0553·23-s − 0.179·25-s − 1.04·27-s − 1.44·29-s − 0.181·31-s − 0.942·33-s − 0.520·35-s + 1.42·37-s + 0.0214·39-s + 1.00·41-s − 1.17·43-s − 0.0822·45-s − 1.00·47-s − 0.669·49-s + 1.20·51-s − 0.627·53-s − 0.895·55-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  =  1
Selberg data  =  $(2,\ 6008,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 + T \)
good3 \( 1 - 1.65T + 3T^{2} \)
5 \( 1 - 2.02T + 5T^{2} \)
7 \( 1 + 1.52T + 7T^{2} \)
11 \( 1 + 3.27T + 11T^{2} \)
13 \( 1 - 0.0809T + 13T^{2} \)
17 \( 1 - 5.20T + 17T^{2} \)
19 \( 1 + 4.81T + 19T^{2} \)
23 \( 1 + 0.265T + 23T^{2} \)
29 \( 1 + 7.75T + 29T^{2} \)
31 \( 1 + 1.01T + 31T^{2} \)
37 \( 1 - 8.64T + 37T^{2} \)
41 \( 1 - 6.42T + 41T^{2} \)
43 \( 1 + 7.71T + 43T^{2} \)
47 \( 1 + 6.87T + 47T^{2} \)
53 \( 1 + 4.57T + 53T^{2} \)
59 \( 1 - 3.22T + 59T^{2} \)
61 \( 1 + 3.60T + 61T^{2} \)
67 \( 1 - 5.12T + 67T^{2} \)
71 \( 1 + 8.35T + 71T^{2} \)
73 \( 1 - 1.23T + 73T^{2} \)
79 \( 1 + 12.3T + 79T^{2} \)
83 \( 1 + 4.33T + 83T^{2} \)
89 \( 1 - 6.77T + 89T^{2} \)
97 \( 1 + 5.92T + 97T^{2} \)
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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

−7.901094048018627433503962454459, −7.16874913681365090817629554266, −6.04709971830199288688874487144, −5.82467147011028913246385076020, −4.88419607985880356927698528019, −3.80494084743687298708882469200, −3.08433210778971737463299306349, −2.42011635003844549588891661372, −1.64904024548769352971055232378, 0, 1.64904024548769352971055232378, 2.42011635003844549588891661372, 3.08433210778971737463299306349, 3.80494084743687298708882469200, 4.88419607985880356927698528019, 5.82467147011028913246385076020, 6.04709971830199288688874487144, 7.16874913681365090817629554266, 7.901094048018627433503962454459

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