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  − 2.96·3-s + 2.30·5-s − 3.32·7-s + 5.79·9-s + 2.14·11-s + 7.10·13-s − 6.84·15-s + 1.94·17-s − 1.57·19-s + 9.87·21-s + 1.61·23-s + 0.331·25-s − 8.27·27-s − 0.641·29-s + 9.34·31-s − 6.36·33-s − 7.68·35-s + 3.74·37-s − 21.0·39-s + 6.15·41-s + 7.41·43-s + 13.3·45-s + 8.21·47-s + 4.08·49-s − 5.76·51-s − 3.90·53-s + 4.95·55-s + ⋯
L(s)  = 1  − 1.71·3-s + 1.03·5-s − 1.25·7-s + 1.93·9-s + 0.647·11-s + 1.97·13-s − 1.76·15-s + 0.471·17-s − 0.361·19-s + 2.15·21-s + 0.337·23-s + 0.0662·25-s − 1.59·27-s − 0.119·29-s + 1.67·31-s − 1.10·33-s − 1.29·35-s + 0.614·37-s − 3.37·39-s + 0.960·41-s + 1.13·43-s + 1.99·45-s + 1.19·47-s + 0.583·49-s − 0.807·51-s − 0.536·53-s + 0.668·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.409722055$
$L(\frac12)$  $\approx$  $1.409722055$
$L(\frac{3}{2})$   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 \( 1 + 2.96T + 3T^{2} \)
5 \( 1 - 2.30T + 5T^{2} \)
7 \( 1 + 3.32T + 7T^{2} \)
11 \( 1 - 2.14T + 11T^{2} \)
13 \( 1 - 7.10T + 13T^{2} \)
17 \( 1 - 1.94T + 17T^{2} \)
19 \( 1 + 1.57T + 19T^{2} \)
23 \( 1 - 1.61T + 23T^{2} \)
29 \( 1 + 0.641T + 29T^{2} \)
31 \( 1 - 9.34T + 31T^{2} \)
37 \( 1 - 3.74T + 37T^{2} \)
41 \( 1 - 6.15T + 41T^{2} \)
43 \( 1 - 7.41T + 43T^{2} \)
47 \( 1 - 8.21T + 47T^{2} \)
53 \( 1 + 3.90T + 53T^{2} \)
59 \( 1 + 4.72T + 59T^{2} \)
61 \( 1 + 3.28T + 61T^{2} \)
67 \( 1 + 1.90T + 67T^{2} \)
71 \( 1 + 7.00T + 71T^{2} \)
73 \( 1 - 5.45T + 73T^{2} \)
79 \( 1 - 9.26T + 79T^{2} \)
83 \( 1 + 4.26T + 83T^{2} \)
89 \( 1 + 5.75T + 89T^{2} \)
97 \( 1 + 3.83T + 97T^{2} \)
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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

−8.009419767990626915857729080918, −6.87130117035498300703029950233, −6.41646857026488237039930663091, −5.88267782702193498329338674653, −5.74342641985140707040805918534, −4.49600343626533633150384805777, −3.84919563436450857882490149315, −2.79374294036048440554541714624, −1.41727897833528634900057368148, −0.76805139774369030979964303442, 0.76805139774369030979964303442, 1.41727897833528634900057368148, 2.79374294036048440554541714624, 3.84919563436450857882490149315, 4.49600343626533633150384805777, 5.74342641985140707040805918534, 5.88267782702193498329338674653, 6.41646857026488237039930663091, 6.87130117035498300703029950233, 8.009419767990626915857729080918

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