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  − 2.09·3-s − 2.89·5-s − 1.97·7-s + 1.40·9-s − 4.41·11-s + 5.29·13-s + 6.06·15-s + 4.12·17-s − 5.91·19-s + 4.14·21-s − 4.03·23-s + 3.35·25-s + 3.33·27-s + 2.04·29-s − 9.54·31-s + 9.26·33-s + 5.69·35-s + 3.08·37-s − 11.1·39-s + 4.34·41-s + 1.34·43-s − 4.07·45-s + 7.33·47-s − 3.11·49-s − 8.65·51-s + 7.15·53-s + 12.7·55-s + ⋯
L(s)  = 1  − 1.21·3-s − 1.29·5-s − 0.745·7-s + 0.469·9-s − 1.33·11-s + 1.46·13-s + 1.56·15-s + 0.999·17-s − 1.35·19-s + 0.903·21-s − 0.842·23-s + 0.670·25-s + 0.642·27-s + 0.379·29-s − 1.71·31-s + 1.61·33-s + 0.963·35-s + 0.507·37-s − 1.77·39-s + 0.678·41-s + 0.205·43-s − 0.607·45-s + 1.07·47-s − 0.444·49-s − 1.21·51-s + 0.982·53-s + 1.71·55-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 \( 1 + 2.09T + 3T^{2} \)
5 \( 1 + 2.89T + 5T^{2} \)
7 \( 1 + 1.97T + 7T^{2} \)
11 \( 1 + 4.41T + 11T^{2} \)
13 \( 1 - 5.29T + 13T^{2} \)
17 \( 1 - 4.12T + 17T^{2} \)
19 \( 1 + 5.91T + 19T^{2} \)
23 \( 1 + 4.03T + 23T^{2} \)
29 \( 1 - 2.04T + 29T^{2} \)
31 \( 1 + 9.54T + 31T^{2} \)
37 \( 1 - 3.08T + 37T^{2} \)
41 \( 1 - 4.34T + 41T^{2} \)
43 \( 1 - 1.34T + 43T^{2} \)
47 \( 1 - 7.33T + 47T^{2} \)
53 \( 1 - 7.15T + 53T^{2} \)
59 \( 1 - 7.09T + 59T^{2} \)
61 \( 1 + 2.47T + 61T^{2} \)
67 \( 1 - 15.5T + 67T^{2} \)
71 \( 1 + 10.9T + 71T^{2} \)
73 \( 1 - 8.28T + 73T^{2} \)
79 \( 1 - 7.28T + 79T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 + 5.72T + 89T^{2} \)
97 \( 1 + 3.73T + 97T^{2} \)
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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

−7.75084480779333606549305878377, −6.96386771413067900267583938162, −6.11816541348087092713954129941, −5.75115724328329137078852205474, −4.92200926494484818923123581396, −3.93708397552830770428193812775, −3.54359533704467948071236504169, −2.37861301843431720340927600110, −0.807179065988432061005270006012, 0, 0.807179065988432061005270006012, 2.37861301843431720340927600110, 3.54359533704467948071236504169, 3.93708397552830770428193812775, 4.92200926494484818923123581396, 5.75115724328329137078852205474, 6.11816541348087092713954129941, 6.96386771413067900267583938162, 7.75084480779333606549305878377

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