Properties

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

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  − 2.97·3-s − 0.794·5-s − 3.63·7-s + 5.87·9-s − 0.674·11-s − 1.02·13-s + 2.36·15-s + 3.65·17-s + 3.51·19-s + 10.8·21-s + 4.04·23-s − 4.36·25-s − 8.54·27-s + 2.95·29-s − 3.73·31-s + 2.00·33-s + 2.88·35-s + 3.30·37-s + 3.06·39-s − 12.0·41-s − 1.05·43-s − 4.66·45-s − 6.46·47-s + 6.24·49-s − 10.8·51-s + 2.27·53-s + 0.535·55-s + ⋯
L(s)  = 1  − 1.71·3-s − 0.355·5-s − 1.37·7-s + 1.95·9-s − 0.203·11-s − 0.285·13-s + 0.610·15-s + 0.886·17-s + 0.807·19-s + 2.36·21-s + 0.844·23-s − 0.873·25-s − 1.64·27-s + 0.548·29-s − 0.671·31-s + 0.349·33-s + 0.488·35-s + 0.543·37-s + 0.490·39-s − 1.88·41-s − 0.161·43-s − 0.694·45-s − 0.942·47-s + 0.891·49-s − 1.52·51-s + 0.313·53-s + 0.0721·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $0.4003188875$
$L(\frac12)$  $\approx$  $0.4003188875$
$L(\frac{3}{2})$   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 \( 1 + 2.97T + 3T^{2} \)
5 \( 1 + 0.794T + 5T^{2} \)
7 \( 1 + 3.63T + 7T^{2} \)
11 \( 1 + 0.674T + 11T^{2} \)
13 \( 1 + 1.02T + 13T^{2} \)
17 \( 1 - 3.65T + 17T^{2} \)
19 \( 1 - 3.51T + 19T^{2} \)
23 \( 1 - 4.04T + 23T^{2} \)
29 \( 1 - 2.95T + 29T^{2} \)
31 \( 1 + 3.73T + 31T^{2} \)
37 \( 1 - 3.30T + 37T^{2} \)
41 \( 1 + 12.0T + 41T^{2} \)
43 \( 1 + 1.05T + 43T^{2} \)
47 \( 1 + 6.46T + 47T^{2} \)
53 \( 1 - 2.27T + 53T^{2} \)
59 \( 1 + 14.8T + 59T^{2} \)
61 \( 1 + 6.97T + 61T^{2} \)
67 \( 1 + 12.8T + 67T^{2} \)
71 \( 1 + 3.34T + 71T^{2} \)
73 \( 1 - 13.2T + 73T^{2} \)
79 \( 1 + 15.7T + 79T^{2} \)
83 \( 1 - 8.63T + 83T^{2} \)
89 \( 1 - 8.61T + 89T^{2} \)
97 \( 1 - 13.8T + 97T^{2} \)
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

−7.69326692320980853141625919912, −7.29344797047827709818783292185, −6.41114953367858934596406325085, −6.09148052464899628045691499780, −5.22692970859695059587359946284, −4.74708754322668651028439933307, −3.62703379419490656145549566268, −3.03664676288141334463036181606, −1.47252157291671972227317965118, −0.37904395860689514337657560141, 0.37904395860689514337657560141, 1.47252157291671972227317965118, 3.03664676288141334463036181606, 3.62703379419490656145549566268, 4.74708754322668651028439933307, 5.22692970859695059587359946284, 6.09148052464899628045691499780, 6.41114953367858934596406325085, 7.29344797047827709818783292185, 7.69326692320980853141625919912

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