# Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 67$ Sign $-0.121 + 0.992i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−1.57 − 0.719i)3-s + (−3.36 + 2.91i)7-s + (1.96 + 2.26i)9-s + (1.63 − 2.54i)13-s + (4.23 − 4.89i)19-s + (7.39 − 2.17i)21-s + (−4.20 − 2.70i)25-s + (−1.46 − 4.98i)27-s + (−6.00 − 9.33i)31-s + 12.1·37-s + (−4.40 + 2.83i)39-s + (−12.9 + 1.85i)43-s + (1.82 − 12.6i)49-s + (−10.2 + 4.65i)57-s + (−4.39 − 14.9i)61-s + ⋯
 L(s)  = 1 + (−0.909 − 0.415i)3-s + (−1.27 + 1.10i)7-s + (0.654 + 0.755i)9-s + (0.453 − 0.705i)13-s + (0.972 − 1.12i)19-s + (1.61 − 0.474i)21-s + (−0.841 − 0.540i)25-s + (−0.281 − 0.959i)27-s + (−1.07 − 1.67i)31-s + 1.99·37-s + (−0.705 + 0.453i)39-s + (−1.97 + 0.283i)43-s + (0.260 − 1.81i)49-s + (−1.35 + 0.617i)57-s + (−0.563 − 1.91i)61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.121 + 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$804$$    =    $$2^{2} \cdot 3 \cdot 67$$ $$\varepsilon$$ = $-0.121 + 0.992i$ motivic weight = $$1$$ character : $\chi_{804} (161, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 804,\ (\ :1/2),\ -0.121 + 0.992i)$ $L(1)$ $\approx$ $0.431835 - 0.488089i$ $L(\frac12)$ $\approx$ $0.431835 - 0.488089i$ $L(\frac{3}{2})$ 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 + (1.57 + 0.719i)T$$
67 $$1 + (-5.79 - 5.78i)T$$
good5 $$1 + (4.20 + 2.70i)T^{2}$$
7 $$1 + (3.36 - 2.91i)T + (0.996 - 6.92i)T^{2}$$
11 $$1 + (9.25 + 5.94i)T^{2}$$
13 $$1 + (-1.63 + 2.54i)T + (-5.40 - 11.8i)T^{2}$$
17 $$1 + (16.3 - 4.78i)T^{2}$$
19 $$1 + (-4.23 + 4.89i)T + (-2.70 - 18.8i)T^{2}$$
23 $$1 + (15.0 + 17.3i)T^{2}$$
29 $$1 - 29T^{2}$$
31 $$1 + (6.00 + 9.33i)T + (-12.8 + 28.1i)T^{2}$$
37 $$1 - 12.1T + 37T^{2}$$
41 $$1 + (-39.3 + 11.5i)T^{2}$$
43 $$1 + (12.9 - 1.85i)T + (41.2 - 12.1i)T^{2}$$
47 $$1 + (30.7 + 35.5i)T^{2}$$
53 $$1 + (-50.8 - 14.9i)T^{2}$$
59 $$1 + (-24.5 + 53.6i)T^{2}$$
61 $$1 + (4.39 + 14.9i)T + (-51.3 + 32.9i)T^{2}$$
71 $$1 + (68.1 + 20.0i)T^{2}$$
73 $$1 + (-5.46 + 1.60i)T + (61.4 - 39.4i)T^{2}$$
79 $$1 + (-9.04 + 14.0i)T + (-32.8 - 71.8i)T^{2}$$
83 $$1 + (-69.8 - 44.8i)T^{2}$$
89 $$1 + (58.2 - 67.2i)T^{2}$$
97 $$1 + 6.97iT - 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}