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  + 0.663·3-s − 1.07·5-s + 2.50·7-s − 2.56·9-s + 2.33·11-s + 3.30·13-s − 0.715·15-s − 4.49·17-s − 1.81·19-s + 1.65·21-s + 1.64·23-s − 3.83·25-s − 3.68·27-s − 7.87·29-s − 4.59·31-s + 1.54·33-s − 2.69·35-s − 5.73·37-s + 2.18·39-s + 10.3·41-s + 9.47·43-s + 2.76·45-s + 10.0·47-s − 0.745·49-s − 2.98·51-s − 11.8·53-s − 2.51·55-s + ⋯
L(s)  = 1  + 0.382·3-s − 0.482·5-s + 0.945·7-s − 0.853·9-s + 0.702·11-s + 0.915·13-s − 0.184·15-s − 1.09·17-s − 0.417·19-s + 0.361·21-s + 0.342·23-s − 0.767·25-s − 0.709·27-s − 1.46·29-s − 0.825·31-s + 0.269·33-s − 0.456·35-s − 0.943·37-s + 0.350·39-s + 1.60·41-s + 1.44·43-s + 0.411·45-s + 1.47·47-s − 0.106·49-s − 0.417·51-s − 1.63·53-s − 0.338·55-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 \( 1 - 0.663T + 3T^{2} \)
5 \( 1 + 1.07T + 5T^{2} \)
7 \( 1 - 2.50T + 7T^{2} \)
11 \( 1 - 2.33T + 11T^{2} \)
13 \( 1 - 3.30T + 13T^{2} \)
17 \( 1 + 4.49T + 17T^{2} \)
19 \( 1 + 1.81T + 19T^{2} \)
23 \( 1 - 1.64T + 23T^{2} \)
29 \( 1 + 7.87T + 29T^{2} \)
31 \( 1 + 4.59T + 31T^{2} \)
37 \( 1 + 5.73T + 37T^{2} \)
41 \( 1 - 10.3T + 41T^{2} \)
43 \( 1 - 9.47T + 43T^{2} \)
47 \( 1 - 10.0T + 47T^{2} \)
53 \( 1 + 11.8T + 53T^{2} \)
59 \( 1 + 13.4T + 59T^{2} \)
61 \( 1 + 6.38T + 61T^{2} \)
67 \( 1 - 4.16T + 67T^{2} \)
71 \( 1 + 2.34T + 71T^{2} \)
73 \( 1 + 13.9T + 73T^{2} \)
79 \( 1 + 15.9T + 79T^{2} \)
83 \( 1 - 13.0T + 83T^{2} \)
89 \( 1 + 13.1T + 89T^{2} \)
97 \( 1 + 12.3T + 97T^{2} \)
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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

−7.66410113409785572300072114752, −7.32272725396612651124982780473, −6.08668702076734701323599077721, −5.79937570029804658087375688580, −4.61777300444256312304804109759, −4.05643956509567251278042250386, −3.32821007826360770100693632733, −2.25066960918456567176270586871, −1.47107416985235976015658637648, 0, 1.47107416985235976015658637648, 2.25066960918456567176270586871, 3.32821007826360770100693632733, 4.05643956509567251278042250386, 4.61777300444256312304804109759, 5.79937570029804658087375688580, 6.08668702076734701323599077721, 7.32272725396612651124982780473, 7.66410113409785572300072114752

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