# Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 5 \cdot 67$ Sign $0.921 - 0.387i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 3-s + 5-s + (−1.59 + 2.76i)7-s + 9-s + (2.11 − 3.66i)11-s + (1.09 + 1.88i)13-s − 15-s + (−2.56 − 4.43i)17-s + (−0.325 − 0.564i)19-s + (1.59 − 2.76i)21-s + (4.02 + 6.96i)23-s + 25-s − 27-s + (−3.94 + 6.83i)29-s + (4.23 − 7.34i)31-s + ⋯
 L(s)  = 1 − 0.577·3-s + 0.447·5-s + (−0.602 + 1.04i)7-s + 0.333·9-s + (0.638 − 1.10i)11-s + (0.302 + 0.523i)13-s − 0.258·15-s + (−0.621 − 1.07i)17-s + (−0.0747 − 0.129i)19-s + (0.347 − 0.602i)21-s + (0.838 + 1.45i)23-s + 0.200·25-s − 0.192·27-s + (−0.733 + 1.27i)29-s + (0.761 − 1.31i)31-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4020$$    =    $$2^{2} \cdot 3 \cdot 5 \cdot 67$$ $$\varepsilon$$ = $0.921 - 0.387i$ motivic weight = $$1$$ character : $\chi_{4020} (841, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 4020,\ (\ :1/2),\ 0.921 - 0.387i)$ $L(1)$ $\approx$ $1.562949375$ $L(\frac12)$ $\approx$ $1.562949375$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;5,\;67\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;67\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
3 $$1 + T$$
5 $$1 - T$$
67 $$1 + (-5.59 + 5.97i)T$$
good7 $$1 + (1.59 - 2.76i)T + (-3.5 - 6.06i)T^{2}$$
11 $$1 + (-2.11 + 3.66i)T + (-5.5 - 9.52i)T^{2}$$
13 $$1 + (-1.09 - 1.88i)T + (-6.5 + 11.2i)T^{2}$$
17 $$1 + (2.56 + 4.43i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (0.325 + 0.564i)T + (-9.5 + 16.4i)T^{2}$$
23 $$1 + (-4.02 - 6.96i)T + (-11.5 + 19.9i)T^{2}$$
29 $$1 + (3.94 - 6.83i)T + (-14.5 - 25.1i)T^{2}$$
31 $$1 + (-4.23 + 7.34i)T + (-15.5 - 26.8i)T^{2}$$
37 $$1 + (1.75 + 3.03i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + (-0.135 + 0.234i)T + (-20.5 - 35.5i)T^{2}$$
43 $$1 - 8.32T + 43T^{2}$$
47 $$1 + (0.171 - 0.297i)T + (-23.5 - 40.7i)T^{2}$$
53 $$1 - 7.89T + 53T^{2}$$
59 $$1 + 4.39T + 59T^{2}$$
61 $$1 + (7.05 + 12.2i)T + (-30.5 + 52.8i)T^{2}$$
71 $$1 + (-1.45 + 2.51i)T + (-35.5 - 61.4i)T^{2}$$
73 $$1 + (-3.18 - 5.51i)T + (-36.5 + 63.2i)T^{2}$$
79 $$1 + (-0.598 + 1.03i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (-0.963 - 1.66i)T + (-41.5 + 71.8i)T^{2}$$
89 $$1 + 3.23T + 89T^{2}$$
97 $$1 + (-5.58 - 9.67i)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}