Properties

Degree 2
Conductor $ 2^{3} \cdot 751 $
Sign $-1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 1

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.94·3-s + 0.760·5-s − 0.889·7-s + 0.765·9-s + 5.58·11-s − 2.54·13-s + 1.47·15-s − 2.48·17-s − 7.62·19-s − 1.72·21-s − 5.94·23-s − 4.42·25-s − 4.33·27-s − 0.846·29-s + 4.09·31-s + 10.8·33-s − 0.676·35-s − 1.60·37-s − 4.93·39-s − 3.27·41-s − 5.80·43-s + 0.582·45-s − 5.13·47-s − 6.20·49-s − 4.82·51-s − 3.26·53-s + 4.25·55-s + ⋯
L(s)  = 1  + 1.12·3-s + 0.340·5-s − 0.336·7-s + 0.255·9-s + 1.68·11-s − 0.705·13-s + 0.381·15-s − 0.603·17-s − 1.74·19-s − 0.376·21-s − 1.23·23-s − 0.884·25-s − 0.834·27-s − 0.157·29-s + 0.734·31-s + 1.88·33-s − 0.114·35-s − 0.264·37-s − 0.790·39-s − 0.511·41-s − 0.885·43-s + 0.0867·45-s − 0.749·47-s − 0.887·49-s − 0.675·51-s − 0.448·53-s + 0.573·55-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  =  1
Selberg data  =  $(2,\ 6008,\ (\ :1/2),\ -1)$
$L(1)$  $=$  $0$
$L(\frac12)$  $=$  $0$
$L(\frac{3}{2})$   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 $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 \( 1 \)
751 \( 1 + T \)
good3 \( 1 - 1.94T + 3T^{2} \)
5 \( 1 - 0.760T + 5T^{2} \)
7 \( 1 + 0.889T + 7T^{2} \)
11 \( 1 - 5.58T + 11T^{2} \)
13 \( 1 + 2.54T + 13T^{2} \)
17 \( 1 + 2.48T + 17T^{2} \)
19 \( 1 + 7.62T + 19T^{2} \)
23 \( 1 + 5.94T + 23T^{2} \)
29 \( 1 + 0.846T + 29T^{2} \)
31 \( 1 - 4.09T + 31T^{2} \)
37 \( 1 + 1.60T + 37T^{2} \)
41 \( 1 + 3.27T + 41T^{2} \)
43 \( 1 + 5.80T + 43T^{2} \)
47 \( 1 + 5.13T + 47T^{2} \)
53 \( 1 + 3.26T + 53T^{2} \)
59 \( 1 - 2.72T + 59T^{2} \)
61 \( 1 - 0.913T + 61T^{2} \)
67 \( 1 - 8.89T + 67T^{2} \)
71 \( 1 + 0.635T + 71T^{2} \)
73 \( 1 - 1.47T + 73T^{2} \)
79 \( 1 - 9.39T + 79T^{2} \)
83 \( 1 + 5.77T + 83T^{2} \)
89 \( 1 + 11.2T + 89T^{2} \)
97 \( 1 + 1.62T + 97T^{2} \)
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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

−8.057642560476106693426228182714, −6.82790593975118898814737220511, −6.53157253834021938062846063182, −5.74203075973733873881018905377, −4.52486226876463752828063870192, −3.97119799071636595023009351664, −3.24843584877539346065749978719, −2.18518812185894711208171778223, −1.77752899763861398306415459932, 0, 1.77752899763861398306415459932, 2.18518812185894711208171778223, 3.24843584877539346065749978719, 3.97119799071636595023009351664, 4.52486226876463752828063870192, 5.74203075973733873881018905377, 6.53157253834021938062846063182, 6.82790593975118898814737220511, 8.057642560476106693426228182714

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