# Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 67$ Sign $-0.737 + 0.675i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.35 − 0.418i)2-s − 3-s + (1.64 + 1.13i)4-s − 1.44i·5-s + (1.35 + 0.418i)6-s + 2.69·7-s + (−1.75 − 2.21i)8-s + 9-s + (−0.606 + 1.95i)10-s − 3.96·11-s + (−1.64 − 1.13i)12-s − 2.95i·13-s + (−3.64 − 1.13i)14-s + 1.44i·15-s + (1.44 + 3.73i)16-s − 2.16·17-s + ⋯
 L(s)  = 1 + (−0.955 − 0.296i)2-s − 0.577·3-s + (0.824 + 0.565i)4-s − 0.647i·5-s + (0.551 + 0.170i)6-s + 1.01·7-s + (−0.620 − 0.784i)8-s + 0.333·9-s + (−0.191 + 0.618i)10-s − 1.19·11-s + (−0.476 − 0.326i)12-s − 0.818i·13-s + (−0.974 − 0.302i)14-s + 0.374i·15-s + (0.360 + 0.932i)16-s − 0.524·17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.737 + 0.675i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.737 + 0.675i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$804$$    =    $$2^{2} \cdot 3 \cdot 67$$ $$\varepsilon$$ = $-0.737 + 0.675i$ motivic weight = $$1$$ character : $\chi_{804} (535, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 804,\ (\ :1/2),\ -0.737 + 0.675i)$ $L(1)$ $\approx$ $0.205018 - 0.527438i$ $L(\frac12)$ $\approx$ $0.205018 - 0.527438i$ $L(\frac{3}{2})$ 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 + (1.35 + 0.418i)T$$
3 $$1 + T$$
67 $$1 + (-1.85 + 7.97i)T$$
good5 $$1 + 1.44iT - 5T^{2}$$
7 $$1 - 2.69T + 7T^{2}$$
11 $$1 + 3.96T + 11T^{2}$$
13 $$1 + 2.95iT - 13T^{2}$$
17 $$1 + 2.16T + 17T^{2}$$
19 $$1 + 6.52iT - 19T^{2}$$
23 $$1 - 5.27iT - 23T^{2}$$
29 $$1 - 1.48T + 29T^{2}$$
31 $$1 - 4.41T + 31T^{2}$$
37 $$1 + 6.67T + 37T^{2}$$
41 $$1 + 0.555iT - 41T^{2}$$
43 $$1 + 10.5T + 43T^{2}$$
47 $$1 + 10.0iT - 47T^{2}$$
53 $$1 - 3.34iT - 53T^{2}$$
59 $$1 + 7.97iT - 59T^{2}$$
61 $$1 + 0.843iT - 61T^{2}$$
71 $$1 + 13.4iT - 71T^{2}$$
73 $$1 + 8.46T + 73T^{2}$$
79 $$1 + 12.5T + 79T^{2}$$
83 $$1 + 8.13iT - 83T^{2}$$
89 $$1 - 3.50T + 89T^{2}$$
97 $$1 + 8.55iT - 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}