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  + 1.91·3-s + 3.07·5-s − 0.591·7-s + 0.673·9-s + 1.99·11-s + 3.60·13-s + 5.89·15-s + 2.69·17-s − 1.18·19-s − 1.13·21-s − 8.54·23-s + 4.44·25-s − 4.45·27-s + 6.20·29-s − 7.60·31-s + 3.81·33-s − 1.81·35-s + 10.3·37-s + 6.91·39-s + 8.86·41-s + 6.29·43-s + 2.07·45-s + 1.22·47-s − 6.64·49-s + 5.16·51-s + 8.63·53-s + 6.11·55-s + ⋯
L(s)  = 1  + 1.10·3-s + 1.37·5-s − 0.223·7-s + 0.224·9-s + 0.600·11-s + 1.00·13-s + 1.52·15-s + 0.653·17-s − 0.272·19-s − 0.247·21-s − 1.78·23-s + 0.888·25-s − 0.858·27-s + 1.15·29-s − 1.36·31-s + 0.664·33-s − 0.307·35-s + 1.69·37-s + 1.10·39-s + 1.38·41-s + 0.960·43-s + 0.308·45-s + 0.179·47-s − 0.949·49-s + 0.722·51-s + 1.18·53-s + 0.824·55-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  =  $(2,\ 6008,\ (\ :1/2),\ 1)$
$L(1)$  $\approx$  $4.232921896$
$L(\frac12)$  $\approx$  $4.232921896$
$L(\frac{3}{2})$   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 \( 1 - 1.91T + 3T^{2} \)
5 \( 1 - 3.07T + 5T^{2} \)
7 \( 1 + 0.591T + 7T^{2} \)
11 \( 1 - 1.99T + 11T^{2} \)
13 \( 1 - 3.60T + 13T^{2} \)
17 \( 1 - 2.69T + 17T^{2} \)
19 \( 1 + 1.18T + 19T^{2} \)
23 \( 1 + 8.54T + 23T^{2} \)
29 \( 1 - 6.20T + 29T^{2} \)
31 \( 1 + 7.60T + 31T^{2} \)
37 \( 1 - 10.3T + 37T^{2} \)
41 \( 1 - 8.86T + 41T^{2} \)
43 \( 1 - 6.29T + 43T^{2} \)
47 \( 1 - 1.22T + 47T^{2} \)
53 \( 1 - 8.63T + 53T^{2} \)
59 \( 1 - 3.53T + 59T^{2} \)
61 \( 1 + 7.16T + 61T^{2} \)
67 \( 1 + 0.813T + 67T^{2} \)
71 \( 1 - 12.1T + 71T^{2} \)
73 \( 1 - 10.8T + 73T^{2} \)
79 \( 1 - 13.0T + 79T^{2} \)
83 \( 1 - 0.290T + 83T^{2} \)
89 \( 1 + 9.51T + 89T^{2} \)
97 \( 1 - 8.90T + 97T^{2} \)
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

−8.148427695085820608594193439681, −7.59292005591894245899367424534, −6.41210966692948824228608073176, −6.08030655339228525349768718554, −5.42874632408269754735825810936, −4.14545050692402616964158416612, −3.62666125855181574866744977451, −2.59893376445593150377833636543, −2.06617535905145384842559359527, −1.09036341323575997288384514950, 1.09036341323575997288384514950, 2.06617535905145384842559359527, 2.59893376445593150377833636543, 3.62666125855181574866744977451, 4.14545050692402616964158416612, 5.42874632408269754735825810936, 6.08030655339228525349768718554, 6.41210966692948824228608073176, 7.59292005591894245899367424534, 8.148427695085820608594193439681

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