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  − 0.410·3-s + 0.180·5-s − 3.49·7-s − 2.83·9-s − 2.01·11-s − 5.49·13-s − 0.0739·15-s − 3.92·17-s − 4.57·19-s + 1.43·21-s + 0.277·23-s − 4.96·25-s + 2.39·27-s − 3.41·29-s − 5.41·31-s + 0.826·33-s − 0.630·35-s − 2.83·37-s + 2.25·39-s + 2.24·41-s − 0.371·43-s − 0.510·45-s + 3.31·47-s + 5.19·49-s + 1.60·51-s + 13.6·53-s − 0.363·55-s + ⋯
L(s)  = 1  − 0.236·3-s + 0.0806·5-s − 1.32·7-s − 0.943·9-s − 0.607·11-s − 1.52·13-s − 0.0191·15-s − 0.951·17-s − 1.04·19-s + 0.312·21-s + 0.0578·23-s − 0.993·25-s + 0.460·27-s − 0.634·29-s − 0.972·31-s + 0.143·33-s − 0.106·35-s − 0.465·37-s + 0.360·39-s + 0.350·41-s − 0.0565·43-s − 0.0761·45-s + 0.483·47-s + 0.742·49-s + 0.225·51-s + 1.87·53-s − 0.0490·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.1054325227$
$L(\frac12)$  $\approx$  $0.1054325227$
$L(\frac{3}{2})$   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 \( 1 + 0.410T + 3T^{2} \)
5 \( 1 - 0.180T + 5T^{2} \)
7 \( 1 + 3.49T + 7T^{2} \)
11 \( 1 + 2.01T + 11T^{2} \)
13 \( 1 + 5.49T + 13T^{2} \)
17 \( 1 + 3.92T + 17T^{2} \)
19 \( 1 + 4.57T + 19T^{2} \)
23 \( 1 - 0.277T + 23T^{2} \)
29 \( 1 + 3.41T + 29T^{2} \)
31 \( 1 + 5.41T + 31T^{2} \)
37 \( 1 + 2.83T + 37T^{2} \)
41 \( 1 - 2.24T + 41T^{2} \)
43 \( 1 + 0.371T + 43T^{2} \)
47 \( 1 - 3.31T + 47T^{2} \)
53 \( 1 - 13.6T + 53T^{2} \)
59 \( 1 - 0.754T + 59T^{2} \)
61 \( 1 + 1.25T + 61T^{2} \)
67 \( 1 + 4.39T + 67T^{2} \)
71 \( 1 + 5.03T + 71T^{2} \)
73 \( 1 + 0.365T + 73T^{2} \)
79 \( 1 + 0.408T + 79T^{2} \)
83 \( 1 + 7.26T + 83T^{2} \)
89 \( 1 - 8.52T + 89T^{2} \)
97 \( 1 - 11.6T + 97T^{2} \)
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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

−8.036308490666839313086963571427, −7.24572933845403992364206397916, −6.68871065605823688304142862139, −5.87393547769047365793644464881, −5.40218954088715952230013715143, −4.44856925518732296725268969156, −3.60434760768765381084042900375, −2.63420188759952011621326998978, −2.18028388974857803523493564924, −0.15560182174009175956941893694, 0.15560182174009175956941893694, 2.18028388974857803523493564924, 2.63420188759952011621326998978, 3.60434760768765381084042900375, 4.44856925518732296725268969156, 5.40218954088715952230013715143, 5.87393547769047365793644464881, 6.68871065605823688304142862139, 7.24572933845403992364206397916, 8.036308490666839313086963571427

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