Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 67$ Sign $0.577 + 0.816i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 − 1.41i·2-s + (−1.41 + i)3-s − 2.00·4-s + 1.41i·5-s + (1.41 + 2.00i)6-s − 2i·7-s + 2.82i·8-s + (1.00 − 2.82i)9-s + 2.00·10-s − 5.65·11-s + (2.82 − 2.00i)12-s + 2·13-s − 2.82·14-s + (−1.41 − 2.00i)15-s + 4.00·16-s + 5.65i·17-s + ⋯
 L(s)  = 1 − 0.999i·2-s + (−0.816 + 0.577i)3-s − 1.00·4-s + 0.632i·5-s + (0.577 + 0.816i)6-s − 0.755i·7-s + 1.00i·8-s + (0.333 − 0.942i)9-s + 0.632·10-s − 1.70·11-s + (0.816 − 0.577i)12-s + 0.554·13-s − 0.755·14-s + (−0.365 − 0.516i)15-s + 1.00·16-s + 1.37i·17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$804$$    =    $$2^{2} \cdot 3 \cdot 67$$ $$\varepsilon$$ = $0.577 + 0.816i$ motivic weight = $$1$$ character : $\chi_{804} (671, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 804,\ (\ :1/2),\ 0.577 + 0.816i)$ $L(1)$ $\approx$ $0.799048 - 0.413617i$ $L(\frac12)$ $\approx$ $0.799048 - 0.413617i$ $L(\frac{3}{2})$ not available $L(1)$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;67\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;67\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + 1.41iT$$
3 $$1 + (1.41 - i)T$$
67 $$1 + iT$$
good5 $$1 - 1.41iT - 5T^{2}$$
7 $$1 + 2iT - 7T^{2}$$
11 $$1 + 5.65T + 11T^{2}$$
13 $$1 - 2T + 13T^{2}$$
17 $$1 - 5.65iT - 17T^{2}$$
19 $$1 + 4iT - 19T^{2}$$
23 $$1 - 7.07T + 23T^{2}$$
29 $$1 + 5.65iT - 29T^{2}$$
31 $$1 - 31T^{2}$$
37 $$1 - 8T + 37T^{2}$$
41 $$1 + 1.41iT - 41T^{2}$$
43 $$1 - 4iT - 43T^{2}$$
47 $$1 - 7.07T + 47T^{2}$$
53 $$1 + 12.7iT - 53T^{2}$$
59 $$1 - 1.41T + 59T^{2}$$
61 $$1 + 6T + 61T^{2}$$
71 $$1 - 15.5T + 71T^{2}$$
73 $$1 - 10T + 73T^{2}$$
79 $$1 + 8iT - 79T^{2}$$
83 $$1 + 15.5T + 83T^{2}$$
89 $$1 - 16.9iT - 89T^{2}$$
97 $$1 + 6T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}