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  + 1.13·3-s − 1.95·5-s − 2.91·7-s − 1.71·9-s + 0.469·11-s − 0.568·13-s − 2.21·15-s + 0.0179·17-s − 5.38·19-s − 3.29·21-s + 1.38·23-s − 1.16·25-s − 5.34·27-s − 1.74·29-s + 2.46·31-s + 0.531·33-s + 5.70·35-s + 4.93·37-s − 0.643·39-s − 0.996·41-s + 3.10·43-s + 3.36·45-s − 11.0·47-s + 1.47·49-s + 0.0203·51-s + 8.32·53-s − 0.919·55-s + ⋯
L(s)  = 1  + 0.653·3-s − 0.876·5-s − 1.10·7-s − 0.572·9-s + 0.141·11-s − 0.157·13-s − 0.572·15-s + 0.00435·17-s − 1.23·19-s − 0.719·21-s + 0.288·23-s − 0.232·25-s − 1.02·27-s − 0.323·29-s + 0.443·31-s + 0.0925·33-s + 0.964·35-s + 0.810·37-s − 0.103·39-s − 0.155·41-s + 0.472·43-s + 0.501·45-s − 1.60·47-s + 0.211·49-s + 0.00284·51-s + 1.14·53-s − 0.123·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9962996078$
$L(\frac12)$  $\approx$  $0.9962996078$
$L(\frac{3}{2})$   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 \( 1 - 1.13T + 3T^{2} \)
5 \( 1 + 1.95T + 5T^{2} \)
7 \( 1 + 2.91T + 7T^{2} \)
11 \( 1 - 0.469T + 11T^{2} \)
13 \( 1 + 0.568T + 13T^{2} \)
17 \( 1 - 0.0179T + 17T^{2} \)
19 \( 1 + 5.38T + 19T^{2} \)
23 \( 1 - 1.38T + 23T^{2} \)
29 \( 1 + 1.74T + 29T^{2} \)
31 \( 1 - 2.46T + 31T^{2} \)
37 \( 1 - 4.93T + 37T^{2} \)
41 \( 1 + 0.996T + 41T^{2} \)
43 \( 1 - 3.10T + 43T^{2} \)
47 \( 1 + 11.0T + 47T^{2} \)
53 \( 1 - 8.32T + 53T^{2} \)
59 \( 1 - 8.95T + 59T^{2} \)
61 \( 1 + 5.75T + 61T^{2} \)
67 \( 1 + 4.44T + 67T^{2} \)
71 \( 1 - 12.6T + 71T^{2} \)
73 \( 1 - 6.04T + 73T^{2} \)
79 \( 1 - 6.75T + 79T^{2} \)
83 \( 1 - 6.31T + 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + 18.4T + 97T^{2} \)
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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

−8.146676195111245130297875888712, −7.50985846735702496684699810851, −6.63365253823671786042156625518, −6.15475524805376735683422823170, −5.16878334523846608651662609831, −4.15313780765041100400691279165, −3.61661886032498321162948648411, −2.89489953155450071384952633905, −2.10407811581562685806139285054, −0.47657268867342001984958095273, 0.47657268867342001984958095273, 2.10407811581562685806139285054, 2.89489953155450071384952633905, 3.61661886032498321162948648411, 4.15313780765041100400691279165, 5.16878334523846608651662609831, 6.15475524805376735683422823170, 6.63365253823671786042156625518, 7.50985846735702496684699810851, 8.146676195111245130297875888712

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