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  − 2.98·3-s + 3.05·5-s + 0.297·7-s + 5.88·9-s − 0.132·11-s + 3.60·13-s − 9.11·15-s + 6.86·17-s − 2.04·19-s − 0.886·21-s − 8.92·23-s + 4.35·25-s − 8.60·27-s − 4.43·29-s − 2.21·31-s + 0.393·33-s + 0.909·35-s − 5.90·37-s − 10.7·39-s − 9.20·41-s + 0.709·43-s + 18.0·45-s + 0.369·47-s − 6.91·49-s − 20.4·51-s − 11.1·53-s − 0.404·55-s + ⋯
L(s)  = 1  − 1.72·3-s + 1.36·5-s + 0.112·7-s + 1.96·9-s − 0.0398·11-s + 0.998·13-s − 2.35·15-s + 1.66·17-s − 0.470·19-s − 0.193·21-s − 1.86·23-s + 0.871·25-s − 1.65·27-s − 0.823·29-s − 0.398·31-s + 0.0685·33-s + 0.153·35-s − 0.971·37-s − 1.71·39-s − 1.43·41-s + 0.108·43-s + 2.68·45-s + 0.0539·47-s − 0.987·49-s − 2.86·51-s − 1.52·53-s − 0.0544·55-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 \( 1 + 2.98T + 3T^{2} \)
5 \( 1 - 3.05T + 5T^{2} \)
7 \( 1 - 0.297T + 7T^{2} \)
11 \( 1 + 0.132T + 11T^{2} \)
13 \( 1 - 3.60T + 13T^{2} \)
17 \( 1 - 6.86T + 17T^{2} \)
19 \( 1 + 2.04T + 19T^{2} \)
23 \( 1 + 8.92T + 23T^{2} \)
29 \( 1 + 4.43T + 29T^{2} \)
31 \( 1 + 2.21T + 31T^{2} \)
37 \( 1 + 5.90T + 37T^{2} \)
41 \( 1 + 9.20T + 41T^{2} \)
43 \( 1 - 0.709T + 43T^{2} \)
47 \( 1 - 0.369T + 47T^{2} \)
53 \( 1 + 11.1T + 53T^{2} \)
59 \( 1 - 5.23T + 59T^{2} \)
61 \( 1 - 0.397T + 61T^{2} \)
67 \( 1 + 4.96T + 67T^{2} \)
71 \( 1 + 3.02T + 71T^{2} \)
73 \( 1 + 9.74T + 73T^{2} \)
79 \( 1 + 4.16T + 79T^{2} \)
83 \( 1 - 4.18T + 83T^{2} \)
89 \( 1 - 10.9T + 89T^{2} \)
97 \( 1 + 2.82T + 97T^{2} \)
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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

−7.60265959876999914523861873845, −6.59076278986399732397305405162, −6.20556862444341394291974016682, −5.53706416508314851534502196432, −5.31643894081438639726012818248, −4.23763925825832995260297689556, −3.35412272685526009519448234923, −1.82381034526752167072436783625, −1.39960338404800342451312364416, 0, 1.39960338404800342451312364416, 1.82381034526752167072436783625, 3.35412272685526009519448234923, 4.23763925825832995260297689556, 5.31643894081438639726012818248, 5.53706416508314851534502196432, 6.20556862444341394291974016682, 6.59076278986399732397305405162, 7.60265959876999914523861873845

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