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  − 2.26·3-s − 1.66·5-s − 0.979·7-s + 2.12·9-s + 5.43·11-s + 0.178·13-s + 3.76·15-s + 5.72·17-s − 3.83·19-s + 2.21·21-s + 2.87·23-s − 2.23·25-s + 1.97·27-s + 8.38·29-s + 3.31·31-s − 12.2·33-s + 1.62·35-s − 0.842·37-s − 0.405·39-s + 1.22·41-s − 12.7·43-s − 3.53·45-s + 1.06·47-s − 6.04·49-s − 12.9·51-s + 8.96·53-s − 9.02·55-s + ⋯
L(s)  = 1  − 1.30·3-s − 0.743·5-s − 0.370·7-s + 0.708·9-s + 1.63·11-s + 0.0496·13-s + 0.971·15-s + 1.38·17-s − 0.879·19-s + 0.483·21-s + 0.599·23-s − 0.447·25-s + 0.380·27-s + 1.55·29-s + 0.594·31-s − 2.14·33-s + 0.275·35-s − 0.138·37-s − 0.0648·39-s + 0.191·41-s − 1.93·43-s − 0.526·45-s + 0.155·47-s − 0.862·49-s − 1.81·51-s + 1.23·53-s − 1.21·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9977078645$
$L(\frac12)$  $\approx$  $0.9977078645$
$L(\frac{3}{2})$   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 \( 1 + 2.26T + 3T^{2} \)
5 \( 1 + 1.66T + 5T^{2} \)
7 \( 1 + 0.979T + 7T^{2} \)
11 \( 1 - 5.43T + 11T^{2} \)
13 \( 1 - 0.178T + 13T^{2} \)
17 \( 1 - 5.72T + 17T^{2} \)
19 \( 1 + 3.83T + 19T^{2} \)
23 \( 1 - 2.87T + 23T^{2} \)
29 \( 1 - 8.38T + 29T^{2} \)
31 \( 1 - 3.31T + 31T^{2} \)
37 \( 1 + 0.842T + 37T^{2} \)
41 \( 1 - 1.22T + 41T^{2} \)
43 \( 1 + 12.7T + 43T^{2} \)
47 \( 1 - 1.06T + 47T^{2} \)
53 \( 1 - 8.96T + 53T^{2} \)
59 \( 1 - 6.92T + 59T^{2} \)
61 \( 1 + 4.91T + 61T^{2} \)
67 \( 1 - 11.8T + 67T^{2} \)
71 \( 1 + 11.2T + 71T^{2} \)
73 \( 1 - 3.04T + 73T^{2} \)
79 \( 1 + 8.00T + 79T^{2} \)
83 \( 1 - 1.01T + 83T^{2} \)
89 \( 1 + 2.43T + 89T^{2} \)
97 \( 1 - 8.02T + 97T^{2} \)
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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

−8.157697892439800427806995353337, −7.03627340450522460047910082215, −6.63662136477522117467027360739, −6.05425820578171052817665931230, −5.25720037071224457938272873617, −4.46585088084004334866983846971, −3.80836398362506790165435351038, −2.98919631908922725890540299296, −1.44266180085531955855873253074, −0.61498207979807457641489623108, 0.61498207979807457641489623108, 1.44266180085531955855873253074, 2.98919631908922725890540299296, 3.80836398362506790165435351038, 4.46585088084004334866983846971, 5.25720037071224457938272873617, 6.05425820578171052817665931230, 6.63662136477522117467027360739, 7.03627340450522460047910082215, 8.157697892439800427806995353337

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