# Properties

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

# Related objects

## Dirichlet series

 L(s)  = 1 + 3.42·3-s + 1.19·5-s − 1.06·7-s + 8.73·9-s + 0.730·11-s + 5.49·13-s + 4.08·15-s + 0.933·17-s + 5.09·19-s − 3.63·21-s − 6.73·23-s − 3.58·25-s + 19.6·27-s + 5.95·29-s + 7.12·31-s + 2.50·33-s − 1.26·35-s + 7.45·37-s + 18.8·39-s − 11.8·41-s − 12.0·43-s + 10.4·45-s − 11.0·47-s − 5.87·49-s + 3.19·51-s − 7.95·53-s + 0.870·55-s + ⋯
 L(s)  = 1 + 1.97·3-s + 0.532·5-s − 0.400·7-s + 2.91·9-s + 0.220·11-s + 1.52·13-s + 1.05·15-s + 0.226·17-s + 1.16·19-s − 0.792·21-s − 1.40·23-s − 0.716·25-s + 3.78·27-s + 1.10·29-s + 1.27·31-s + 0.435·33-s − 0.213·35-s + 1.22·37-s + 3.01·39-s − 1.85·41-s − 1.83·43-s + 1.55·45-s − 1.61·47-s − 0.839·49-s + 0.447·51-s − 1.09·53-s + 0.117·55-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\n
\begin{aligned} \Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\n

## 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 = $(2,\ 6008,\ (\ :1/2),\ 1)$ $L(1)$ $\approx$ $5.447262838$ $L(\frac12)$ $\approx$ $5.447262838$ $L(\frac{3}{2})$ 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 - 3.42T + 3T^{2}$$
5 $$1 - 1.19T + 5T^{2}$$
7 $$1 + 1.06T + 7T^{2}$$
11 $$1 - 0.730T + 11T^{2}$$
13 $$1 - 5.49T + 13T^{2}$$
17 $$1 - 0.933T + 17T^{2}$$
19 $$1 - 5.09T + 19T^{2}$$
23 $$1 + 6.73T + 23T^{2}$$
29 $$1 - 5.95T + 29T^{2}$$
31 $$1 - 7.12T + 31T^{2}$$
37 $$1 - 7.45T + 37T^{2}$$
41 $$1 + 11.8T + 41T^{2}$$
43 $$1 + 12.0T + 43T^{2}$$
47 $$1 + 11.0T + 47T^{2}$$
53 $$1 + 7.95T + 53T^{2}$$
59 $$1 + 8.66T + 59T^{2}$$
61 $$1 - 11.6T + 61T^{2}$$
67 $$1 - 0.0992T + 67T^{2}$$
71 $$1 + 0.301T + 71T^{2}$$
73 $$1 - 1.95T + 73T^{2}$$
79 $$1 - 0.440T + 79T^{2}$$
83 $$1 + 10.9T + 83T^{2}$$
89 $$1 - 5.62T + 89T^{2}$$
97 $$1 - 2.36T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}