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  − 0.208·3-s + 0.579·5-s + 3.19·7-s − 2.95·9-s + 3.04·11-s + 2.78·13-s − 0.120·15-s + 1.55·17-s + 4.47·19-s − 0.666·21-s + 8.96·23-s − 4.66·25-s + 1.24·27-s + 5.11·29-s + 4.73·31-s − 0.635·33-s + 1.85·35-s + 1.13·37-s − 0.581·39-s − 7.41·41-s + 6.89·43-s − 1.71·45-s − 8.88·47-s + 3.21·49-s − 0.324·51-s − 10.6·53-s + 1.76·55-s + ⋯
L(s)  = 1  − 0.120·3-s + 0.259·5-s + 1.20·7-s − 0.985·9-s + 0.919·11-s + 0.773·13-s − 0.0312·15-s + 0.377·17-s + 1.02·19-s − 0.145·21-s + 1.86·23-s − 0.932·25-s + 0.238·27-s + 0.949·29-s + 0.850·31-s − 0.110·33-s + 0.313·35-s + 0.186·37-s − 0.0931·39-s − 1.15·41-s + 1.05·43-s − 0.255·45-s − 1.29·47-s + 0.458·49-s − 0.0454·51-s − 1.46·53-s + 0.238·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $2.724702430$
$L(\frac12)$  $\approx$  $2.724702430$
$L(\frac{3}{2})$   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 \( 1 + 0.208T + 3T^{2} \)
5 \( 1 - 0.579T + 5T^{2} \)
7 \( 1 - 3.19T + 7T^{2} \)
11 \( 1 - 3.04T + 11T^{2} \)
13 \( 1 - 2.78T + 13T^{2} \)
17 \( 1 - 1.55T + 17T^{2} \)
19 \( 1 - 4.47T + 19T^{2} \)
23 \( 1 - 8.96T + 23T^{2} \)
29 \( 1 - 5.11T + 29T^{2} \)
31 \( 1 - 4.73T + 31T^{2} \)
37 \( 1 - 1.13T + 37T^{2} \)
41 \( 1 + 7.41T + 41T^{2} \)
43 \( 1 - 6.89T + 43T^{2} \)
47 \( 1 + 8.88T + 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 - 2.39T + 59T^{2} \)
61 \( 1 + 8.89T + 61T^{2} \)
67 \( 1 + 13.7T + 67T^{2} \)
71 \( 1 + 4.02T + 71T^{2} \)
73 \( 1 - 16.4T + 73T^{2} \)
79 \( 1 - 4.37T + 79T^{2} \)
83 \( 1 + 10.2T + 83T^{2} \)
89 \( 1 - 9.54T + 89T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−8.103238236242245832810544306818, −7.50478183314457092303160274273, −6.49506483250992455743873978457, −6.01241365417676182147337728048, −5.08826682299896174370884660137, −4.70138470236431128015484747530, −3.51988762485761381693116196601, −2.88806631830116306870494252934, −1.63848819660613979988479541490, −0.969750086994406510010334684237, 0.969750086994406510010334684237, 1.63848819660613979988479541490, 2.88806631830116306870494252934, 3.51988762485761381693116196601, 4.70138470236431128015484747530, 5.08826682299896174370884660137, 6.01241365417676182147337728048, 6.49506483250992455743873978457, 7.50478183314457092303160274273, 8.103238236242245832810544306818

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