# Properties

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

# Related objects

## 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}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}