# Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 5 \cdot 67$ Sign $-0.996 + 0.0877i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − i·3-s + (−2.22 + 0.196i)5-s + 0.128i·7-s − 9-s + 0.0389·11-s − 3.81i·13-s + (0.196 + 2.22i)15-s + 6.63i·17-s + 7.30·19-s + 0.128·21-s − 3.64i·23-s + (4.92 − 0.874i)25-s + i·27-s − 8.25·29-s − 7.07·31-s + ⋯
 L(s)  = 1 − 0.577i·3-s + (−0.996 + 0.0877i)5-s + 0.0484i·7-s − 0.333·9-s + 0.0117·11-s − 1.05i·13-s + (0.0506 + 0.575i)15-s + 1.61i·17-s + 1.67·19-s + 0.0279·21-s − 0.760i·23-s + (0.984 − 0.174i)25-s + 0.192i·27-s − 1.53·29-s − 1.27·31-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 + 0.0877i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.996 + 0.0877i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4020$$    =    $$2^{2} \cdot 3 \cdot 5 \cdot 67$$ $$\varepsilon$$ = $-0.996 + 0.0877i$ motivic weight = $$1$$ character : $\chi_{4020} (1609, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 4020,\ (\ :1/2),\ -0.996 + 0.0877i)$ $L(1)$ $\approx$ $0.4818936313$ $L(\frac12)$ $\approx$ $0.4818936313$ $L(\frac{3}{2})$ 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 + iT$$
5 $$1 + (2.22 - 0.196i)T$$
67 $$1 + iT$$
good7 $$1 - 0.128iT - 7T^{2}$$
11 $$1 - 0.0389T + 11T^{2}$$
13 $$1 + 3.81iT - 13T^{2}$$
17 $$1 - 6.63iT - 17T^{2}$$
19 $$1 - 7.30T + 19T^{2}$$
23 $$1 + 3.64iT - 23T^{2}$$
29 $$1 + 8.25T + 29T^{2}$$
31 $$1 + 7.07T + 31T^{2}$$
37 $$1 + 7.37iT - 37T^{2}$$
41 $$1 - 1.87T + 41T^{2}$$
43 $$1 + 7.72iT - 43T^{2}$$
47 $$1 - 9.62iT - 47T^{2}$$
53 $$1 + 3.82iT - 53T^{2}$$
59 $$1 - 9.48T + 59T^{2}$$
61 $$1 + 6.69T + 61T^{2}$$
71 $$1 - 4.88T + 71T^{2}$$
73 $$1 + 0.309iT - 73T^{2}$$
79 $$1 + 5.72T + 79T^{2}$$
83 $$1 + 14.6iT - 83T^{2}$$
89 $$1 + 11.6T + 89T^{2}$$
97 $$1 - 7.10iT - 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}