# Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 67$ Sign $-0.713 - 0.700i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.487 + 1.66i)3-s + (1.00 + 1.95i)7-s + (−2.52 − 1.62i)9-s + (4.37 + 5.56i)13-s + (−5.95 − 3.07i)19-s + (−3.74 + 0.720i)21-s + (0.711 + 4.94i)25-s + (3.92 − 3.40i)27-s + (−4.43 + 5.64i)31-s + (4.14 + 7.17i)37-s + (−11.3 + 4.55i)39-s + (−11.4 − 5.24i)43-s + (1.25 − 1.76i)49-s + (8.00 − 8.39i)57-s + (−13.8 − 4.79i)61-s + ⋯
 L(s)  = 1 + (−0.281 + 0.959i)3-s + (0.380 + 0.738i)7-s + (−0.841 − 0.540i)9-s + (1.21 + 1.54i)13-s + (−1.36 − 0.704i)19-s + (−0.816 + 0.157i)21-s + (0.142 + 0.989i)25-s + (0.755 − 0.654i)27-s + (−0.797 + 1.01i)31-s + (0.680 + 1.17i)37-s + (−1.82 + 0.729i)39-s + (−1.75 − 0.799i)43-s + (0.179 − 0.251i)49-s + (1.06 − 1.11i)57-s + (−1.77 − 0.614i)61-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.713 - 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$804$$    =    $$2^{2} \cdot 3 \cdot 67$$ $$\varepsilon$$ = $-0.713 - 0.700i$ motivic weight = $$1$$ character : $\chi_{804} (605, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 804,\ (\ :1/2),\ -0.713 - 0.700i)$ $L(1)$ $\approx$ $0.446702 + 1.09281i$ $L(\frac12)$ $\approx$ $0.446702 + 1.09281i$ $L(\frac{3}{2})$ 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 + (0.487 - 1.66i)T$$
67 $$1 + (-6.78 - 4.58i)T$$
good5 $$1 + (-0.711 - 4.94i)T^{2}$$
7 $$1 + (-1.00 - 1.95i)T + (-4.06 + 5.70i)T^{2}$$
11 $$1 + (-8.64 + 6.79i)T^{2}$$
13 $$1 + (-4.37 - 5.56i)T + (-3.06 + 12.6i)T^{2}$$
17 $$1 + (-16.6 + 3.21i)T^{2}$$
19 $$1 + (5.95 + 3.07i)T + (11.0 + 15.4i)T^{2}$$
23 $$1 + (20.4 - 10.5i)T^{2}$$
29 $$1 + (14.5 + 25.1i)T^{2}$$
31 $$1 + (4.43 - 5.64i)T + (-7.30 - 30.1i)T^{2}$$
37 $$1 + (-4.14 - 7.17i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + (-13.4 + 38.7i)T^{2}$$
43 $$1 + (11.4 + 5.24i)T + (28.1 + 32.4i)T^{2}$$
47 $$1 + (-2.23 + 46.9i)T^{2}$$
53 $$1 + (-34.7 + 40.0i)T^{2}$$
59 $$1 + (56.6 + 16.6i)T^{2}$$
61 $$1 + (13.8 + 4.79i)T + (47.9 + 37.7i)T^{2}$$
71 $$1 + (-69.7 - 13.4i)T^{2}$$
73 $$1 + (2.35 - 6.80i)T + (-57.3 - 45.1i)T^{2}$$
79 $$1 + (4.14 - 10.3i)T + (-57.1 - 54.5i)T^{2}$$
83 $$1 + (-77.0 - 30.8i)T^{2}$$
89 $$1 + (-74.8 + 48.1i)T^{2}$$
97 $$1 + (-14.6 + 8.43i)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}