Properties

Degree 2
Conductor $ 2^{3} \cdot 751 $
Sign $1$
Motivic weight 1
Primitive yes
Self-dual yes
Analytic rank 0

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.13·3-s + 3.35·5-s + 0.313·7-s + 1.54·9-s + 2.50·11-s − 6.16·13-s + 7.15·15-s + 5.24·17-s + 7.03·19-s + 0.668·21-s + 1.85·23-s + 6.26·25-s − 3.09·27-s − 7.13·29-s + 8.61·31-s + 5.33·33-s + 1.05·35-s + 11.0·37-s − 13.1·39-s − 5.42·41-s − 4.95·43-s + 5.19·45-s + 2.99·47-s − 6.90·49-s + 11.1·51-s + 6.00·53-s + 8.40·55-s + ⋯
L(s)  = 1  + 1.23·3-s + 1.50·5-s + 0.118·7-s + 0.515·9-s + 0.754·11-s − 1.70·13-s + 1.84·15-s + 1.27·17-s + 1.61·19-s + 0.145·21-s + 0.386·23-s + 1.25·25-s − 0.596·27-s − 1.32·29-s + 1.54·31-s + 0.928·33-s + 0.177·35-s + 1.82·37-s − 2.10·39-s − 0.847·41-s − 0.755·43-s + 0.774·45-s + 0.437·47-s − 0.985·49-s + 1.56·51-s + 0.824·53-s + 1.13·55-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  =  0
Selberg data  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.628991165$
$L(\frac12)$  $\approx$  $4.628991165$
$L(\frac{3}{2})$   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 $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 \( 1 \)
751 \( 1 + T \)
good3 \( 1 - 2.13T + 3T^{2} \)
5 \( 1 - 3.35T + 5T^{2} \)
7 \( 1 - 0.313T + 7T^{2} \)
11 \( 1 - 2.50T + 11T^{2} \)
13 \( 1 + 6.16T + 13T^{2} \)
17 \( 1 - 5.24T + 17T^{2} \)
19 \( 1 - 7.03T + 19T^{2} \)
23 \( 1 - 1.85T + 23T^{2} \)
29 \( 1 + 7.13T + 29T^{2} \)
31 \( 1 - 8.61T + 31T^{2} \)
37 \( 1 - 11.0T + 37T^{2} \)
41 \( 1 + 5.42T + 41T^{2} \)
43 \( 1 + 4.95T + 43T^{2} \)
47 \( 1 - 2.99T + 47T^{2} \)
53 \( 1 - 6.00T + 53T^{2} \)
59 \( 1 - 10.4T + 59T^{2} \)
61 \( 1 + 12.3T + 61T^{2} \)
67 \( 1 + 7.81T + 67T^{2} \)
71 \( 1 + 2.57T + 71T^{2} \)
73 \( 1 + 4.30T + 73T^{2} \)
79 \( 1 - 2.00T + 79T^{2} \)
83 \( 1 + 12.5T + 83T^{2} \)
89 \( 1 - 3.54T + 89T^{2} \)
97 \( 1 - 9.22T + 97T^{2} \)
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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

−7.963069713753299722359951379556, −7.55186807325996190493379815179, −6.78390660229616750852487898810, −5.81799438084774776380545455411, −5.31468119137836747166612646607, −4.46196290418315271587163033125, −3.26549021739605237849334848818, −2.80538175497198135397225448669, −1.98857203290337473965402847209, −1.16064380610540900113912333287, 1.16064380610540900113912333287, 1.98857203290337473965402847209, 2.80538175497198135397225448669, 3.26549021739605237849334848818, 4.46196290418315271587163033125, 5.31468119137836747166612646607, 5.81799438084774776380545455411, 6.78390660229616750852487898810, 7.55186807325996190493379815179, 7.963069713753299722359951379556

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