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  + 2.62·3-s − 0.385·5-s + 1.06·7-s + 3.91·9-s − 3.06·11-s − 4.43·13-s − 1.01·15-s − 2.91·17-s − 0.422·19-s + 2.78·21-s − 0.921·23-s − 4.85·25-s + 2.39·27-s − 8.01·29-s + 6.77·31-s − 8.06·33-s − 0.409·35-s + 4.94·37-s − 11.6·39-s + 7.67·41-s + 1.55·43-s − 1.50·45-s − 4.43·47-s − 5.87·49-s − 7.67·51-s − 9.11·53-s + 1.18·55-s + ⋯
L(s)  = 1  + 1.51·3-s − 0.172·5-s + 0.400·7-s + 1.30·9-s − 0.924·11-s − 1.23·13-s − 0.261·15-s − 0.707·17-s − 0.0968·19-s + 0.608·21-s − 0.192·23-s − 0.970·25-s + 0.460·27-s − 1.48·29-s + 1.21·31-s − 1.40·33-s − 0.0691·35-s + 0.812·37-s − 1.86·39-s + 1.19·41-s + 0.237·43-s − 0.225·45-s − 0.647·47-s − 0.839·49-s − 1.07·51-s − 1.25·53-s + 0.159·55-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 \( 1 - 2.62T + 3T^{2} \)
5 \( 1 + 0.385T + 5T^{2} \)
7 \( 1 - 1.06T + 7T^{2} \)
11 \( 1 + 3.06T + 11T^{2} \)
13 \( 1 + 4.43T + 13T^{2} \)
17 \( 1 + 2.91T + 17T^{2} \)
19 \( 1 + 0.422T + 19T^{2} \)
23 \( 1 + 0.921T + 23T^{2} \)
29 \( 1 + 8.01T + 29T^{2} \)
31 \( 1 - 6.77T + 31T^{2} \)
37 \( 1 - 4.94T + 37T^{2} \)
41 \( 1 - 7.67T + 41T^{2} \)
43 \( 1 - 1.55T + 43T^{2} \)
47 \( 1 + 4.43T + 47T^{2} \)
53 \( 1 + 9.11T + 53T^{2} \)
59 \( 1 - 1.46T + 59T^{2} \)
61 \( 1 - 2.75T + 61T^{2} \)
67 \( 1 + 11.8T + 67T^{2} \)
71 \( 1 - 8.47T + 71T^{2} \)
73 \( 1 + 14.0T + 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 - 13.7T + 83T^{2} \)
89 \( 1 - 2.11T + 89T^{2} \)
97 \( 1 - 1.01T + 97T^{2} \)
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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

−7.76205406943421812073404127945, −7.48849257954793272950671112342, −6.44244616788502112976584449207, −5.46970094795678525658720282526, −4.59479367704574985513795090350, −4.04349347496327145165085174565, −2.98794237175640436094389672849, −2.45767573965225957352217390402, −1.73437748517393853595401009380, 0, 1.73437748517393853595401009380, 2.45767573965225957352217390402, 2.98794237175640436094389672849, 4.04349347496327145165085174565, 4.59479367704574985513795090350, 5.46970094795678525658720282526, 6.44244616788502112976584449207, 7.48849257954793272950671112342, 7.76205406943421812073404127945

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