# Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 67$ Sign $-0.860 + 0.509i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1 − 1.41i)3-s − 2·5-s + (−0.949 − 0.548i)7-s + (−1.00 − 2.82i)9-s + (0.724 − 1.25i)11-s + (4.5 − 2.59i)13-s + (−2 + 2.82i)15-s + (−5.17 + 2.98i)17-s + (−1.5 − 2.59i)19-s + (−1.72 + 0.794i)21-s + (−0.825 + 0.476i)23-s − 25-s + (−5.00 − 1.41i)27-s + (−4.62 − 2.66i)29-s + (−1.5 − 0.866i)31-s + ⋯
 L(s)  = 1 + (0.577 − 0.816i)3-s − 0.894·5-s + (−0.358 − 0.207i)7-s + (−0.333 − 0.942i)9-s + (0.218 − 0.378i)11-s + (1.24 − 0.720i)13-s + (−0.516 + 0.730i)15-s + (−1.25 + 0.724i)17-s + (−0.344 − 0.596i)19-s + (−0.376 + 0.173i)21-s + (−0.172 + 0.0994i)23-s − 0.200·25-s + (−0.962 − 0.272i)27-s + (−0.858 − 0.495i)29-s + (−0.269 − 0.155i)31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$804$$    =    $$2^{2} \cdot 3 \cdot 67$$ $$\varepsilon$$ = $-0.860 + 0.509i$ motivic weight = $$1$$ character : $\chi_{804} (365, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 804,\ (\ :1/2),\ -0.860 + 0.509i)$ $L(1)$ $\approx$ $0.273049 - 0.995930i$ $L(\frac12)$ $\approx$ $0.273049 - 0.995930i$ $L(\frac{3}{2})$ 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$$
3 $$1 + (-1 + 1.41i)T$$
67 $$1 + (-8 - 1.73i)T$$
good5 $$1 + 2T + 5T^{2}$$
7 $$1 + (0.949 + 0.548i)T + (3.5 + 6.06i)T^{2}$$
11 $$1 + (-0.724 + 1.25i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + (-4.5 + 2.59i)T + (6.5 - 11.2i)T^{2}$$
17 $$1 + (5.17 - 2.98i)T + (8.5 - 14.7i)T^{2}$$
19 $$1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (0.825 - 0.476i)T + (11.5 - 19.9i)T^{2}$$
29 $$1 + (4.62 + 2.66i)T + (14.5 + 25.1i)T^{2}$$
31 $$1 + (1.5 + 0.866i)T + (15.5 + 26.8i)T^{2}$$
37 $$1 + (4.94 + 8.57i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + (1.72 - 2.98i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 - 1.27iT - 43T^{2}$$
47 $$1 + (0.275 + 0.158i)T + (23.5 + 40.7i)T^{2}$$
53 $$1 - 6T + 53T^{2}$$
59 $$1 + 0.635iT - 59T^{2}$$
61 $$1 + (-9.39 + 5.42i)T + (30.5 - 52.8i)T^{2}$$
71 $$1 + (-11.1 - 6.45i)T + (35.5 + 61.4i)T^{2}$$
73 $$1 + (-2.94 - 5.10i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (-0.398 - 0.230i)T + (39.5 + 68.4i)T^{2}$$
83 $$1 + (10.0 - 5.81i)T + (41.5 - 71.8i)T^{2}$$
89 $$1 + 4.73iT - 89T^{2}$$
97 $$1 + (-11.8 + 6.84i)T + (48.5 - 84.0i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}