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  − 1.61·3-s − 2.17·5-s − 2.82·7-s − 0.382·9-s − 0.331·11-s + 1.18·13-s + 3.52·15-s − 0.850·17-s − 2.54·19-s + 4.57·21-s − 2.89·23-s − 0.247·25-s + 5.47·27-s + 3.27·29-s + 9.17·31-s + 0.536·33-s + 6.16·35-s + 4.19·37-s − 1.91·39-s + 10.8·41-s + 5.52·43-s + 0.834·45-s − 3.07·47-s + 0.999·49-s + 1.37·51-s − 7.07·53-s + 0.722·55-s + ⋯
L(s)  = 1  − 0.934·3-s − 0.974·5-s − 1.06·7-s − 0.127·9-s − 0.0999·11-s + 0.328·13-s + 0.910·15-s − 0.206·17-s − 0.584·19-s + 0.998·21-s − 0.603·23-s − 0.0495·25-s + 1.05·27-s + 0.607·29-s + 1.64·31-s + 0.0933·33-s + 1.04·35-s + 0.690·37-s − 0.307·39-s + 1.69·41-s + 0.842·43-s + 0.124·45-s − 0.448·47-s + 0.142·49-s + 0.192·51-s − 0.971·53-s + 0.0974·55-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 \( 1 + 1.61T + 3T^{2} \)
5 \( 1 + 2.17T + 5T^{2} \)
7 \( 1 + 2.82T + 7T^{2} \)
11 \( 1 + 0.331T + 11T^{2} \)
13 \( 1 - 1.18T + 13T^{2} \)
17 \( 1 + 0.850T + 17T^{2} \)
19 \( 1 + 2.54T + 19T^{2} \)
23 \( 1 + 2.89T + 23T^{2} \)
29 \( 1 - 3.27T + 29T^{2} \)
31 \( 1 - 9.17T + 31T^{2} \)
37 \( 1 - 4.19T + 37T^{2} \)
41 \( 1 - 10.8T + 41T^{2} \)
43 \( 1 - 5.52T + 43T^{2} \)
47 \( 1 + 3.07T + 47T^{2} \)
53 \( 1 + 7.07T + 53T^{2} \)
59 \( 1 + 5.25T + 59T^{2} \)
61 \( 1 - 5.14T + 61T^{2} \)
67 \( 1 + 12.7T + 67T^{2} \)
71 \( 1 - 6.59T + 71T^{2} \)
73 \( 1 - 8.22T + 73T^{2} \)
79 \( 1 + 7.17T + 79T^{2} \)
83 \( 1 - 5.27T + 83T^{2} \)
89 \( 1 - 15.0T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−7.81313671748260087993791720026, −6.80037453661749934400219852867, −6.24497242192591418755750415821, −5.84524864855425231108815883650, −4.69729082234488344977416185487, −4.19178967154015858325618998899, −3.27010697460738037974518014994, −2.50150411536056274674387432664, −0.868026396193306314343871014461, 0, 0.868026396193306314343871014461, 2.50150411536056274674387432664, 3.27010697460738037974518014994, 4.19178967154015858325618998899, 4.69729082234488344977416185487, 5.84524864855425231108815883650, 6.24497242192591418755750415821, 6.80037453661749934400219852867, 7.81313671748260087993791720026

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