# Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 5 \cdot 67$ Sign $0.930 + 0.365i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + i·3-s + (2.08 + 0.817i)5-s − 3.41i·7-s − 9-s + 5.18·11-s + 5.24i·13-s + (−0.817 + 2.08i)15-s − 7.93i·17-s + 1.59·19-s + 3.41·21-s − 7.16i·23-s + (3.66 + 3.40i)25-s − i·27-s + 6.58·29-s − 5.96·31-s + ⋯
 L(s)  = 1 + 0.577i·3-s + (0.930 + 0.365i)5-s − 1.29i·7-s − 0.333·9-s + 1.56·11-s + 1.45i·13-s + (−0.211 + 0.537i)15-s − 1.92i·17-s + 0.366·19-s + 0.745·21-s − 1.49i·23-s + (0.732 + 0.680i)25-s − 0.192i·27-s + 1.22·29-s − 1.07·31-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.930 + 0.365i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4020$$    =    $$2^{2} \cdot 3 \cdot 5 \cdot 67$$ $$\varepsilon$$ = $0.930 + 0.365i$ motivic weight = $$1$$ character : $\chi_{4020} (1609, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 4020,\ (\ :1/2),\ 0.930 + 0.365i)$ $L(1)$ $\approx$ $2.518755626$ $L(\frac12)$ $\approx$ $2.518755626$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;67\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;67\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1 - iT$$
5 $$1 + (-2.08 - 0.817i)T$$
67 $$1 - iT$$
good7 $$1 + 3.41iT - 7T^{2}$$
11 $$1 - 5.18T + 11T^{2}$$
13 $$1 - 5.24iT - 13T^{2}$$
17 $$1 + 7.93iT - 17T^{2}$$
19 $$1 - 1.59T + 19T^{2}$$
23 $$1 + 7.16iT - 23T^{2}$$
29 $$1 - 6.58T + 29T^{2}$$
31 $$1 + 5.96T + 31T^{2}$$
37 $$1 + 10.2iT - 37T^{2}$$
41 $$1 + 8.07T + 41T^{2}$$
43 $$1 + 0.663iT - 43T^{2}$$
47 $$1 + 7.08iT - 47T^{2}$$
53 $$1 - 11.5iT - 53T^{2}$$
59 $$1 - 1.90T + 59T^{2}$$
61 $$1 - 1.14T + 61T^{2}$$
71 $$1 - 1.39T + 71T^{2}$$
73 $$1 - 6.22iT - 73T^{2}$$
79 $$1 + 1.75T + 79T^{2}$$
83 $$1 + 12.9iT - 83T^{2}$$
89 $$1 + 6.29T + 89T^{2}$$
97 $$1 + 4.62iT - 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}