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  − 3.18·3-s − 2.41·5-s + 2.14·7-s + 7.12·9-s + 4.27·11-s + 6.04·13-s + 7.67·15-s − 4.10·17-s − 5.68·19-s − 6.82·21-s − 7.79·23-s + 0.817·25-s − 13.1·27-s + 4.70·29-s + 2.59·31-s − 13.6·33-s − 5.17·35-s − 9.09·37-s − 19.2·39-s + 0.788·41-s + 8.65·43-s − 17.1·45-s − 3.21·47-s − 2.40·49-s + 13.0·51-s + 14.2·53-s − 10.3·55-s + ⋯
L(s)  = 1  − 1.83·3-s − 1.07·5-s + 0.810·7-s + 2.37·9-s + 1.29·11-s + 1.67·13-s + 1.98·15-s − 0.994·17-s − 1.30·19-s − 1.48·21-s − 1.62·23-s + 0.163·25-s − 2.52·27-s + 0.874·29-s + 0.466·31-s − 2.37·33-s − 0.874·35-s − 1.49·37-s − 3.07·39-s + 0.123·41-s + 1.32·43-s − 2.56·45-s − 0.468·47-s − 0.343·49-s + 1.82·51-s + 1.96·53-s − 1.39·55-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 \( 1 + 3.18T + 3T^{2} \)
5 \( 1 + 2.41T + 5T^{2} \)
7 \( 1 - 2.14T + 7T^{2} \)
11 \( 1 - 4.27T + 11T^{2} \)
13 \( 1 - 6.04T + 13T^{2} \)
17 \( 1 + 4.10T + 17T^{2} \)
19 \( 1 + 5.68T + 19T^{2} \)
23 \( 1 + 7.79T + 23T^{2} \)
29 \( 1 - 4.70T + 29T^{2} \)
31 \( 1 - 2.59T + 31T^{2} \)
37 \( 1 + 9.09T + 37T^{2} \)
41 \( 1 - 0.788T + 41T^{2} \)
43 \( 1 - 8.65T + 43T^{2} \)
47 \( 1 + 3.21T + 47T^{2} \)
53 \( 1 - 14.2T + 53T^{2} \)
59 \( 1 - 1.45T + 59T^{2} \)
61 \( 1 + 3.31T + 61T^{2} \)
67 \( 1 + 8.27T + 67T^{2} \)
71 \( 1 - 6.59T + 71T^{2} \)
73 \( 1 + 8.46T + 73T^{2} \)
79 \( 1 + 5.68T + 79T^{2} \)
83 \( 1 + 2.17T + 83T^{2} \)
89 \( 1 + 3.16T + 89T^{2} \)
97 \( 1 - 13.7T + 97T^{2} \)
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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

−7.59949126186875108040531953404, −6.73907000290709334696350464647, −6.30327383343151771100140001219, −5.76985167487427473840720223065, −4.67480980302965512467039220595, −4.13583384574779985333774092043, −3.84466855841185376910291589898, −1.89319602626267788525637164286, −1.07926071039763555709813272107, 0, 1.07926071039763555709813272107, 1.89319602626267788525637164286, 3.84466855841185376910291589898, 4.13583384574779985333774092043, 4.67480980302965512467039220595, 5.76985167487427473840720223065, 6.30327383343151771100140001219, 6.73907000290709334696350464647, 7.59949126186875108040531953404

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