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  − 2.12·3-s − 1.44·5-s + 4.49·7-s + 1.49·9-s + 6.23·11-s − 4.90·13-s + 3.06·15-s − 4.00·17-s + 4.28·19-s − 9.54·21-s − 0.204·23-s − 2.90·25-s + 3.18·27-s + 2.81·29-s − 2.53·31-s − 13.2·33-s − 6.50·35-s − 4.71·37-s + 10.4·39-s − 5.20·41-s − 5.82·43-s − 2.16·45-s − 9.40·47-s + 13.2·49-s + 8.49·51-s − 6.67·53-s − 9.01·55-s + ⋯
L(s)  = 1  − 1.22·3-s − 0.647·5-s + 1.70·7-s + 0.499·9-s + 1.87·11-s − 1.36·13-s + 0.792·15-s − 0.971·17-s + 0.982·19-s − 2.08·21-s − 0.0426·23-s − 0.581·25-s + 0.612·27-s + 0.523·29-s − 0.455·31-s − 2.30·33-s − 1.10·35-s − 0.775·37-s + 1.66·39-s − 0.812·41-s − 0.888·43-s − 0.323·45-s − 1.37·47-s + 1.89·49-s + 1.18·51-s − 0.916·53-s − 1.21·55-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 \( 1 + 2.12T + 3T^{2} \)
5 \( 1 + 1.44T + 5T^{2} \)
7 \( 1 - 4.49T + 7T^{2} \)
11 \( 1 - 6.23T + 11T^{2} \)
13 \( 1 + 4.90T + 13T^{2} \)
17 \( 1 + 4.00T + 17T^{2} \)
19 \( 1 - 4.28T + 19T^{2} \)
23 \( 1 + 0.204T + 23T^{2} \)
29 \( 1 - 2.81T + 29T^{2} \)
31 \( 1 + 2.53T + 31T^{2} \)
37 \( 1 + 4.71T + 37T^{2} \)
41 \( 1 + 5.20T + 41T^{2} \)
43 \( 1 + 5.82T + 43T^{2} \)
47 \( 1 + 9.40T + 47T^{2} \)
53 \( 1 + 6.67T + 53T^{2} \)
59 \( 1 + 8.26T + 59T^{2} \)
61 \( 1 + 3.12T + 61T^{2} \)
67 \( 1 - 6.50T + 67T^{2} \)
71 \( 1 - 10.0T + 71T^{2} \)
73 \( 1 - 3.29T + 73T^{2} \)
79 \( 1 - 1.95T + 79T^{2} \)
83 \( 1 - 15.5T + 83T^{2} \)
89 \( 1 - 0.548T + 89T^{2} \)
97 \( 1 + 18.6T + 97T^{2} \)
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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

−7.70598867174431594814811272174, −6.85220120267800312048393959857, −6.47145720262741679945090494207, −5.32719072174775069553163338423, −4.89517997990883490115143557768, −4.36039959468957563228235122473, −3.44694300477940876495057467213, −1.99318935096517425242985051719, −1.23745233800560637422609383925, 0, 1.23745233800560637422609383925, 1.99318935096517425242985051719, 3.44694300477940876495057467213, 4.36039959468957563228235122473, 4.89517997990883490115143557768, 5.32719072174775069553163338423, 6.47145720262741679945090494207, 6.85220120267800312048393959857, 7.70598867174431594814811272174

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