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  − 0.504·3-s − 0.903·5-s + 1.64·7-s − 2.74·9-s + 2.24·11-s − 6.33·13-s + 0.456·15-s − 3.81·17-s − 2.50·19-s − 0.829·21-s − 4.38·23-s − 4.18·25-s + 2.89·27-s − 1.88·29-s + 9.57·31-s − 1.13·33-s − 1.48·35-s + 8.55·37-s + 3.19·39-s + 2.33·41-s − 0.990·43-s + 2.48·45-s − 3.32·47-s − 4.29·49-s + 1.92·51-s − 6.47·53-s − 2.02·55-s + ⋯
L(s)  = 1  − 0.291·3-s − 0.404·5-s + 0.621·7-s − 0.915·9-s + 0.675·11-s − 1.75·13-s + 0.117·15-s − 0.924·17-s − 0.573·19-s − 0.181·21-s − 0.913·23-s − 0.836·25-s + 0.557·27-s − 0.350·29-s + 1.71·31-s − 0.196·33-s − 0.251·35-s + 1.40·37-s + 0.511·39-s + 0.363·41-s − 0.150·43-s + 0.369·45-s − 0.484·47-s − 0.613·49-s + 0.269·51-s − 0.888·53-s − 0.272·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.9513828980$
$L(\frac12)$  $\approx$  $0.9513828980$
$L(\frac{3}{2})$   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 \( 1 + 0.504T + 3T^{2} \)
5 \( 1 + 0.903T + 5T^{2} \)
7 \( 1 - 1.64T + 7T^{2} \)
11 \( 1 - 2.24T + 11T^{2} \)
13 \( 1 + 6.33T + 13T^{2} \)
17 \( 1 + 3.81T + 17T^{2} \)
19 \( 1 + 2.50T + 19T^{2} \)
23 \( 1 + 4.38T + 23T^{2} \)
29 \( 1 + 1.88T + 29T^{2} \)
31 \( 1 - 9.57T + 31T^{2} \)
37 \( 1 - 8.55T + 37T^{2} \)
41 \( 1 - 2.33T + 41T^{2} \)
43 \( 1 + 0.990T + 43T^{2} \)
47 \( 1 + 3.32T + 47T^{2} \)
53 \( 1 + 6.47T + 53T^{2} \)
59 \( 1 - 3.39T + 59T^{2} \)
61 \( 1 + 9.94T + 61T^{2} \)
67 \( 1 - 11.7T + 67T^{2} \)
71 \( 1 - 9.71T + 71T^{2} \)
73 \( 1 - 3.32T + 73T^{2} \)
79 \( 1 - 5.95T + 79T^{2} \)
83 \( 1 - 9.44T + 83T^{2} \)
89 \( 1 + 1.70T + 89T^{2} \)
97 \( 1 - 10.7T + 97T^{2} \)
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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

−8.006966918274628440786134235484, −7.55062919229947371624935928606, −6.46161768941176180712759678819, −6.15173513907205163120152539524, −4.97291448384866574908810704529, −4.62877121363245624296036625907, −3.76768557149446210715979830548, −2.61648868623433576426177391516, −2.00569020692458625983908092899, −0.49470184620429845199188203372, 0.49470184620429845199188203372, 2.00569020692458625983908092899, 2.61648868623433576426177391516, 3.76768557149446210715979830548, 4.62877121363245624296036625907, 4.97291448384866574908810704529, 6.15173513907205163120152539524, 6.46161768941176180712759678819, 7.55062919229947371624935928606, 8.006966918274628440786134235484

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