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  − 3.25·3-s + 3.98·5-s − 3.42·7-s + 7.59·9-s + 3.25·11-s − 0.807·13-s − 12.9·15-s − 7.09·17-s − 1.26·19-s + 11.1·21-s + 4.19·23-s + 10.8·25-s − 14.9·27-s + 9.60·29-s − 3.56·31-s − 10.6·33-s − 13.6·35-s − 8.53·37-s + 2.62·39-s + 1.78·41-s − 3.07·43-s + 30.2·45-s + 1.74·47-s + 4.70·49-s + 23.0·51-s − 8.29·53-s + 12.9·55-s + ⋯
L(s)  = 1  − 1.87·3-s + 1.78·5-s − 1.29·7-s + 2.53·9-s + 0.982·11-s − 0.223·13-s − 3.34·15-s − 1.72·17-s − 0.289·19-s + 2.43·21-s + 0.873·23-s + 2.17·25-s − 2.88·27-s + 1.78·29-s − 0.639·31-s − 1.84·33-s − 2.30·35-s − 1.40·37-s + 0.420·39-s + 0.279·41-s − 0.468·43-s + 4.51·45-s + 0.254·47-s + 0.671·49-s + 3.23·51-s − 1.13·53-s + 1.74·55-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 \( 1 + 3.25T + 3T^{2} \)
5 \( 1 - 3.98T + 5T^{2} \)
7 \( 1 + 3.42T + 7T^{2} \)
11 \( 1 - 3.25T + 11T^{2} \)
13 \( 1 + 0.807T + 13T^{2} \)
17 \( 1 + 7.09T + 17T^{2} \)
19 \( 1 + 1.26T + 19T^{2} \)
23 \( 1 - 4.19T + 23T^{2} \)
29 \( 1 - 9.60T + 29T^{2} \)
31 \( 1 + 3.56T + 31T^{2} \)
37 \( 1 + 8.53T + 37T^{2} \)
41 \( 1 - 1.78T + 41T^{2} \)
43 \( 1 + 3.07T + 43T^{2} \)
47 \( 1 - 1.74T + 47T^{2} \)
53 \( 1 + 8.29T + 53T^{2} \)
59 \( 1 + 8.47T + 59T^{2} \)
61 \( 1 + 5.19T + 61T^{2} \)
67 \( 1 + 4.78T + 67T^{2} \)
71 \( 1 + 3.44T + 71T^{2} \)
73 \( 1 - 7.82T + 73T^{2} \)
79 \( 1 + 10.4T + 79T^{2} \)
83 \( 1 + 6.41T + 83T^{2} \)
89 \( 1 + 0.0939T + 89T^{2} \)
97 \( 1 - 7.80T + 97T^{2} \)
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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

−7.04900126902498109922963640768, −6.71793692585879894267374256772, −6.26551340033387474526685269369, −5.86768772121500784854004627799, −4.92168895055204625966405821886, −4.48586063473541427606354224321, −3.19170028687472227181128938437, −2.04483496226397049258866266056, −1.19885081152647687614009897262, 0, 1.19885081152647687614009897262, 2.04483496226397049258866266056, 3.19170028687472227181128938437, 4.48586063473541427606354224321, 4.92168895055204625966405821886, 5.86768772121500784854004627799, 6.26551340033387474526685269369, 6.71793692585879894267374256772, 7.04900126902498109922963640768

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