Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + 2.96·3-s + 0.884·5-s − 3.59·7-s + 5.79·9-s − 5.74·11-s − 0.730·13-s + 2.62·15-s + 1.55·17-s − 0.000218·19-s − 10.6·21-s + 5.06·23-s − 4.21·25-s + 8.28·27-s − 2.90·29-s + 2.03·31-s − 17.0·33-s − 3.18·35-s − 9.59·37-s − 2.16·39-s − 5.53·41-s − 5.49·43-s + 5.12·45-s − 4.78·47-s + 5.93·49-s + 4.61·51-s + 6.57·53-s − 5.08·55-s + ⋯
L(s)  = 1  + 1.71·3-s + 0.395·5-s − 1.35·7-s + 1.93·9-s − 1.73·11-s − 0.202·13-s + 0.677·15-s + 0.377·17-s − 5.00e − 5·19-s − 2.32·21-s + 1.05·23-s − 0.843·25-s + 1.59·27-s − 0.539·29-s + 0.364·31-s − 2.96·33-s − 0.537·35-s − 1.57·37-s − 0.346·39-s − 0.864·41-s − 0.838·43-s + 0.764·45-s − 0.698·47-s + 0.847·49-s + 0.646·51-s + 0.903·53-s − 0.685·55-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 \( 1 - 2.96T + 3T^{2} \)
5 \( 1 - 0.884T + 5T^{2} \)
7 \( 1 + 3.59T + 7T^{2} \)
11 \( 1 + 5.74T + 11T^{2} \)
13 \( 1 + 0.730T + 13T^{2} \)
17 \( 1 - 1.55T + 17T^{2} \)
19 \( 1 + 0.000218T + 19T^{2} \)
23 \( 1 - 5.06T + 23T^{2} \)
29 \( 1 + 2.90T + 29T^{2} \)
31 \( 1 - 2.03T + 31T^{2} \)
37 \( 1 + 9.59T + 37T^{2} \)
41 \( 1 + 5.53T + 41T^{2} \)
43 \( 1 + 5.49T + 43T^{2} \)
47 \( 1 + 4.78T + 47T^{2} \)
53 \( 1 - 6.57T + 53T^{2} \)
59 \( 1 + 10.1T + 59T^{2} \)
61 \( 1 + 0.100T + 61T^{2} \)
67 \( 1 + 13.6T + 67T^{2} \)
71 \( 1 + 10.7T + 71T^{2} \)
73 \( 1 - 12.8T + 73T^{2} \)
79 \( 1 - 12.1T + 79T^{2} \)
83 \( 1 - 4.78T + 83T^{2} \)
89 \( 1 + 14.9T + 89T^{2} \)
97 \( 1 - 2.87T + 97T^{2} \)
show more
show less
\[\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}\]

Imaginary part of the first few zeros on the critical line

−7.76157650739730330072017754733, −7.24236047630112307396685816818, −6.49385908064027077439016072582, −5.52487748259192721093162848215, −4.79539760951132295818342475749, −3.59048136691348799727660778607, −3.15944635294387467944337490368, −2.55684185148236638351947305385, −1.71643318549574525985467568235, 0, 1.71643318549574525985467568235, 2.55684185148236638351947305385, 3.15944635294387467944337490368, 3.59048136691348799727660778607, 4.79539760951132295818342475749, 5.52487748259192721093162848215, 6.49385908064027077439016072582, 7.24236047630112307396685816818, 7.76157650739730330072017754733

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