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  + 0.325·3-s − 3.38·5-s + 0.980·7-s − 2.89·9-s − 2.93·11-s + 4.83·13-s − 1.10·15-s + 7.26·17-s − 3.24·19-s + 0.319·21-s − 4.67·23-s + 6.46·25-s − 1.91·27-s + 0.705·29-s − 2.40·31-s − 0.953·33-s − 3.32·35-s + 3.15·37-s + 1.57·39-s − 7.63·41-s + 0.860·43-s + 9.80·45-s − 7.21·47-s − 6.03·49-s + 2.36·51-s − 4.24·53-s + 9.92·55-s + ⋯
L(s)  = 1  + 0.187·3-s − 1.51·5-s + 0.370·7-s − 0.964·9-s − 0.883·11-s + 1.34·13-s − 0.284·15-s + 1.76·17-s − 0.745·19-s + 0.0696·21-s − 0.975·23-s + 1.29·25-s − 0.369·27-s + 0.130·29-s − 0.432·31-s − 0.165·33-s − 0.561·35-s + 0.517·37-s + 0.252·39-s − 1.19·41-s + 0.131·43-s + 1.46·45-s − 1.05·47-s − 0.862·49-s + 0.330·51-s − 0.583·53-s + 1.33·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $1.070952329$
$L(\frac12)$  $\approx$  $1.070952329$
$L(\frac{3}{2})$   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 \( 1 - 0.325T + 3T^{2} \)
5 \( 1 + 3.38T + 5T^{2} \)
7 \( 1 - 0.980T + 7T^{2} \)
11 \( 1 + 2.93T + 11T^{2} \)
13 \( 1 - 4.83T + 13T^{2} \)
17 \( 1 - 7.26T + 17T^{2} \)
19 \( 1 + 3.24T + 19T^{2} \)
23 \( 1 + 4.67T + 23T^{2} \)
29 \( 1 - 0.705T + 29T^{2} \)
31 \( 1 + 2.40T + 31T^{2} \)
37 \( 1 - 3.15T + 37T^{2} \)
41 \( 1 + 7.63T + 41T^{2} \)
43 \( 1 - 0.860T + 43T^{2} \)
47 \( 1 + 7.21T + 47T^{2} \)
53 \( 1 + 4.24T + 53T^{2} \)
59 \( 1 + 8.60T + 59T^{2} \)
61 \( 1 - 3.47T + 61T^{2} \)
67 \( 1 - 12.7T + 67T^{2} \)
71 \( 1 + 0.381T + 71T^{2} \)
73 \( 1 - 1.63T + 73T^{2} \)
79 \( 1 + 3.41T + 79T^{2} \)
83 \( 1 - 6.72T + 83T^{2} \)
89 \( 1 - 2.86T + 89T^{2} \)
97 \( 1 - 14.6T + 97T^{2} \)
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

−8.101547170242110239349914624385, −7.80263673601869847688964763249, −6.70170924152185230594525042255, −5.88682427725898398457798068537, −5.21499379166470890726095444038, −4.32790410587961337191825695186, −3.46275223789122028429756146719, −3.16128005199399524513368143428, −1.83042085415230484406889675456, −0.52995835788933192351007613501, 0.52995835788933192351007613501, 1.83042085415230484406889675456, 3.16128005199399524513368143428, 3.46275223789122028429756146719, 4.32790410587961337191825695186, 5.21499379166470890726095444038, 5.88682427725898398457798068537, 6.70170924152185230594525042255, 7.80263673601869847688964763249, 8.101547170242110239349914624385

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