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  − 2.46·3-s + 1.54·5-s − 4.51·7-s + 3.07·9-s − 1.39·11-s − 1.88·13-s − 3.80·15-s + 5.81·17-s − 5.42·19-s + 11.1·21-s + 1.48·23-s − 2.60·25-s − 0.173·27-s + 6.06·29-s + 2.12·31-s + 3.43·33-s − 6.97·35-s + 5.90·37-s + 4.64·39-s + 0.00128·41-s + 1.74·43-s + 4.74·45-s − 4.47·47-s + 13.3·49-s − 14.3·51-s + 1.11·53-s − 2.15·55-s + ⋯
L(s)  = 1  − 1.42·3-s + 0.691·5-s − 1.70·7-s + 1.02·9-s − 0.420·11-s − 0.523·13-s − 0.983·15-s + 1.41·17-s − 1.24·19-s + 2.42·21-s + 0.310·23-s − 0.521·25-s − 0.0333·27-s + 1.12·29-s + 0.381·31-s + 0.598·33-s − 1.17·35-s + 0.970·37-s + 0.744·39-s + 0.000200·41-s + 0.265·43-s + 0.707·45-s − 0.652·47-s + 1.91·49-s − 2.00·51-s + 0.153·53-s − 0.291·55-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 \( 1 + 2.46T + 3T^{2} \)
5 \( 1 - 1.54T + 5T^{2} \)
7 \( 1 + 4.51T + 7T^{2} \)
11 \( 1 + 1.39T + 11T^{2} \)
13 \( 1 + 1.88T + 13T^{2} \)
17 \( 1 - 5.81T + 17T^{2} \)
19 \( 1 + 5.42T + 19T^{2} \)
23 \( 1 - 1.48T + 23T^{2} \)
29 \( 1 - 6.06T + 29T^{2} \)
31 \( 1 - 2.12T + 31T^{2} \)
37 \( 1 - 5.90T + 37T^{2} \)
41 \( 1 - 0.00128T + 41T^{2} \)
43 \( 1 - 1.74T + 43T^{2} \)
47 \( 1 + 4.47T + 47T^{2} \)
53 \( 1 - 1.11T + 53T^{2} \)
59 \( 1 - 3.60T + 59T^{2} \)
61 \( 1 + 7.40T + 61T^{2} \)
67 \( 1 - 10.7T + 67T^{2} \)
71 \( 1 - 6.58T + 71T^{2} \)
73 \( 1 + 8.24T + 73T^{2} \)
79 \( 1 - 7.11T + 79T^{2} \)
83 \( 1 - 3.52T + 83T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 - 15.2T + 97T^{2} \)
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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

−7.51294523100294702184554429368, −6.61146327128646183181682925776, −6.31699415531470185185764487879, −5.71214599038378362852812741970, −5.09643113831397826151409015312, −4.19112949579390235640701452763, −3.15636875342342952440989822504, −2.38886972433666814439220438079, −0.964951951064832553478997929232, 0, 0.964951951064832553478997929232, 2.38886972433666814439220438079, 3.15636875342342952440989822504, 4.19112949579390235640701452763, 5.09643113831397826151409015312, 5.71214599038378362852812741970, 6.31699415531470185185764487879, 6.61146327128646183181682925776, 7.51294523100294702184554429368

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