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  + 1.61·3-s − 2.04·5-s − 3.31·7-s − 0.386·9-s − 2.70·11-s − 2.54·13-s − 3.31·15-s − 6.55·17-s + 7.23·19-s − 5.35·21-s + 2.74·23-s − 0.797·25-s − 5.47·27-s − 5.48·29-s + 7.20·31-s − 4.36·33-s + 6.79·35-s + 3.67·37-s − 4.11·39-s − 8.99·41-s + 11.5·43-s + 0.792·45-s − 1.90·47-s + 3.98·49-s − 10.5·51-s − 3.98·53-s + 5.53·55-s + ⋯
L(s)  = 1  + 0.933·3-s − 0.916·5-s − 1.25·7-s − 0.128·9-s − 0.814·11-s − 0.706·13-s − 0.855·15-s − 1.59·17-s + 1.65·19-s − 1.16·21-s + 0.571·23-s − 0.159·25-s − 1.05·27-s − 1.01·29-s + 1.29·31-s − 0.760·33-s + 1.14·35-s + 0.603·37-s − 0.659·39-s − 1.40·41-s + 1.76·43-s + 0.118·45-s − 0.278·47-s + 0.569·49-s − 1.48·51-s − 0.547·53-s + 0.746·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9655080689$
$L(\frac12)$  $\approx$  $0.9655080689$
$L(\frac{3}{2})$   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 \( 1 - 1.61T + 3T^{2} \)
5 \( 1 + 2.04T + 5T^{2} \)
7 \( 1 + 3.31T + 7T^{2} \)
11 \( 1 + 2.70T + 11T^{2} \)
13 \( 1 + 2.54T + 13T^{2} \)
17 \( 1 + 6.55T + 17T^{2} \)
19 \( 1 - 7.23T + 19T^{2} \)
23 \( 1 - 2.74T + 23T^{2} \)
29 \( 1 + 5.48T + 29T^{2} \)
31 \( 1 - 7.20T + 31T^{2} \)
37 \( 1 - 3.67T + 37T^{2} \)
41 \( 1 + 8.99T + 41T^{2} \)
43 \( 1 - 11.5T + 43T^{2} \)
47 \( 1 + 1.90T + 47T^{2} \)
53 \( 1 + 3.98T + 53T^{2} \)
59 \( 1 + 4.53T + 59T^{2} \)
61 \( 1 - 11.9T + 61T^{2} \)
67 \( 1 + 4.99T + 67T^{2} \)
71 \( 1 - 9.44T + 71T^{2} \)
73 \( 1 - 4.74T + 73T^{2} \)
79 \( 1 - 5.82T + 79T^{2} \)
83 \( 1 + 16.6T + 83T^{2} \)
89 \( 1 + 7.87T + 89T^{2} \)
97 \( 1 - 17.2T + 97T^{2} \)
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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

−8.055750264621789371420515192206, −7.44354260880611311869950990275, −6.92597156991998070308989830794, −6.00042308786609338406672840389, −5.12631868135791354685464561150, −4.26645964697859664689616674132, −3.41376005112730203803927591842, −2.92376970568627202580830597599, −2.22149123701060389251635853123, −0.45652030490516711471773060352, 0.45652030490516711471773060352, 2.22149123701060389251635853123, 2.92376970568627202580830597599, 3.41376005112730203803927591842, 4.26645964697859664689616674132, 5.12631868135791354685464561150, 6.00042308786609338406672840389, 6.92597156991998070308989830794, 7.44354260880611311869950990275, 8.055750264621789371420515192206

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