# Properties

 Degree 2 Conductor $2^{6} \cdot 3 \cdot 7$ Sign $-0.220 - 0.975i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (1 + 1.41i)3-s + (0.414 − 0.414i)5-s + 7-s + (−1.00 + 2.82i)9-s + (3.82 + 3.82i)11-s + (−4.41 + 4.41i)13-s + (1 + 0.171i)15-s − 1.17i·17-s + (−0.414 − 0.414i)19-s + (1 + 1.41i)21-s − 7.65i·23-s + 4.65i·25-s + (−5.00 + 1.41i)27-s + (−3 − 3i)29-s + 6.48i·31-s + ⋯
 L(s)  = 1 + (0.577 + 0.816i)3-s + (0.185 − 0.185i)5-s + 0.377·7-s + (−0.333 + 0.942i)9-s + (1.15 + 1.15i)11-s + (−1.22 + 1.22i)13-s + (0.258 + 0.0442i)15-s − 0.284i·17-s + (−0.0950 − 0.0950i)19-s + (0.218 + 0.308i)21-s − 1.59i·23-s + 0.931i·25-s + (−0.962 + 0.272i)27-s + (−0.557 − 0.557i)29-s + 1.16i·31-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$1344$$    =    $$2^{6} \cdot 3 \cdot 7$$ $$\varepsilon$$ = $-0.220 - 0.975i$ motivic weight = $$1$$ character : $\chi_{1344} (911, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 1344,\ (\ :1/2),\ -0.220 - 0.975i)$ $L(1)$ $\approx$ $2.016198138$ $L(\frac12)$ $\approx$ $2.016198138$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;7\}$, $$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
3 $$1 + (-1 - 1.41i)T$$
7 $$1 - T$$
good5 $$1 + (-0.414 + 0.414i)T - 5iT^{2}$$
11 $$1 + (-3.82 - 3.82i)T + 11iT^{2}$$
13 $$1 + (4.41 - 4.41i)T - 13iT^{2}$$
17 $$1 + 1.17iT - 17T^{2}$$
19 $$1 + (0.414 + 0.414i)T + 19iT^{2}$$
23 $$1 + 7.65iT - 23T^{2}$$
29 $$1 + (3 + 3i)T + 29iT^{2}$$
31 $$1 - 6.48iT - 31T^{2}$$
37 $$1 + (-1.82 - 1.82i)T + 37iT^{2}$$
41 $$1 + 0.343T + 41T^{2}$$
43 $$1 + (-3.82 + 3.82i)T - 43iT^{2}$$
47 $$1 - 5.65T + 47T^{2}$$
53 $$1 + (5.82 - 5.82i)T - 53iT^{2}$$
59 $$1 + (-10.0 - 10.0i)T + 59iT^{2}$$
61 $$1 + (-2.41 + 2.41i)T - 61iT^{2}$$
67 $$1 + (-7 - 7i)T + 67iT^{2}$$
71 $$1 + 6iT - 71T^{2}$$
73 $$1 + 5.17iT - 73T^{2}$$
79 $$1 - 2iT - 79T^{2}$$
83 $$1 + (8.89 - 8.89i)T - 83iT^{2}$$
89 $$1 + 15.6T + 89T^{2}$$
97 $$1 + 2T + 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}