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  + 3.04·3-s + 0.431·5-s − 0.556·7-s + 6.27·9-s + 5.64·11-s − 1.81·13-s + 1.31·15-s − 5.52·17-s − 2.92·19-s − 1.69·21-s + 3.66·23-s − 4.81·25-s + 9.97·27-s + 7.96·29-s + 10.0·31-s + 17.1·33-s − 0.240·35-s − 8.39·37-s − 5.52·39-s − 2.60·41-s + 7.95·43-s + 2.70·45-s + 6.97·47-s − 6.68·49-s − 16.8·51-s + 13.4·53-s + 2.43·55-s + ⋯
L(s)  = 1  + 1.75·3-s + 0.193·5-s − 0.210·7-s + 2.09·9-s + 1.70·11-s − 0.502·13-s + 0.339·15-s − 1.33·17-s − 0.671·19-s − 0.370·21-s + 0.763·23-s − 0.962·25-s + 1.91·27-s + 1.47·29-s + 1.80·31-s + 2.99·33-s − 0.0406·35-s − 1.38·37-s − 0.884·39-s − 0.407·41-s + 1.21·43-s + 0.403·45-s + 1.01·47-s − 0.955·49-s − 2.35·51-s + 1.84·53-s + 0.328·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.480900488$
$L(\frac12)$  $\approx$  $4.480900488$
$L(\frac{3}{2})$   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 \( 1 - 3.04T + 3T^{2} \)
5 \( 1 - 0.431T + 5T^{2} \)
7 \( 1 + 0.556T + 7T^{2} \)
11 \( 1 - 5.64T + 11T^{2} \)
13 \( 1 + 1.81T + 13T^{2} \)
17 \( 1 + 5.52T + 17T^{2} \)
19 \( 1 + 2.92T + 19T^{2} \)
23 \( 1 - 3.66T + 23T^{2} \)
29 \( 1 - 7.96T + 29T^{2} \)
31 \( 1 - 10.0T + 31T^{2} \)
37 \( 1 + 8.39T + 37T^{2} \)
41 \( 1 + 2.60T + 41T^{2} \)
43 \( 1 - 7.95T + 43T^{2} \)
47 \( 1 - 6.97T + 47T^{2} \)
53 \( 1 - 13.4T + 53T^{2} \)
59 \( 1 - 5.82T + 59T^{2} \)
61 \( 1 + 5.35T + 61T^{2} \)
67 \( 1 + 1.25T + 67T^{2} \)
71 \( 1 - 15.4T + 71T^{2} \)
73 \( 1 - 6.63T + 73T^{2} \)
79 \( 1 + 8.96T + 79T^{2} \)
83 \( 1 - 5.14T + 83T^{2} \)
89 \( 1 + 1.35T + 89T^{2} \)
97 \( 1 - 9.11T + 97T^{2} \)
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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

−8.307632525029283396725448404064, −7.42727932251634019087169339107, −6.67293178549977149254349638579, −6.38302626760934568407924653301, −4.89081522103282054889775385421, −4.16620450512400984752259429543, −3.67407156419132057677587629903, −2.64022739516752353747436866935, −2.14815759552317889421882917060, −1.08374480520662128358931729047, 1.08374480520662128358931729047, 2.14815759552317889421882917060, 2.64022739516752353747436866935, 3.67407156419132057677587629903, 4.16620450512400984752259429543, 4.89081522103282054889775385421, 6.38302626760934568407924653301, 6.67293178549977149254349638579, 7.42727932251634019087169339107, 8.307632525029283396725448404064

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