# Properties

 Degree 2 Conductor $2^{6} \cdot 3^{2} \cdot 7$ Sign $-0.691 - 0.722i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.11·5-s + (−1.37 + 2.26i)7-s − 0.812·11-s + 2·13-s + 3.55i·17-s − 4.52i·19-s − 3.63i·23-s − 3.75·25-s + 8.95i·29-s − 5.08·31-s + (−1.53 + 2.52i)35-s + 0.718i·37-s + 10.4i·41-s + 1.11·43-s + 8.55·47-s + ⋯
 L(s)  = 1 + 0.498·5-s + (−0.518 + 0.854i)7-s − 0.245·11-s + 0.554·13-s + 0.862i·17-s − 1.03i·19-s − 0.758i·23-s − 0.751·25-s + 1.66i·29-s − 0.914·31-s + (−0.258 + 0.426i)35-s + 0.118i·37-s + 1.63i·41-s + 0.170·43-s + 1.24·47-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4032$$    =    $$2^{6} \cdot 3^{2} \cdot 7$$ $$\varepsilon$$ = $-0.691 - 0.722i$ motivic weight = $$1$$ character : $\chi_{4032} (1567, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 4032,\ (\ :1/2),\ -0.691 - 0.722i)$ $L(1)$ $\approx$ $1.056216052$ $L(\frac12)$ $\approx$ $1.056216052$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;7\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
3 $$1$$
7 $$1 + (1.37 - 2.26i)T$$
good5 $$1 - 1.11T + 5T^{2}$$
11 $$1 + 0.812T + 11T^{2}$$
13 $$1 - 2T + 13T^{2}$$
17 $$1 - 3.55iT - 17T^{2}$$
19 $$1 + 4.52iT - 19T^{2}$$
23 $$1 + 3.63iT - 23T^{2}$$
29 $$1 - 8.95iT - 29T^{2}$$
31 $$1 + 5.08T + 31T^{2}$$
37 $$1 - 0.718iT - 37T^{2}$$
41 $$1 - 10.4iT - 41T^{2}$$
43 $$1 - 1.11T + 43T^{2}$$
47 $$1 - 8.55T + 47T^{2}$$
53 $$1 - 5.08iT - 53T^{2}$$
59 $$1 - 2.23iT - 59T^{2}$$
61 $$1 - 5.27T + 61T^{2}$$
67 $$1 + 15.1T + 67T^{2}$$
71 $$1 + 5.87iT - 71T^{2}$$
73 $$1 - 3.86iT - 73T^{2}$$
79 $$1 - 1.70iT - 79T^{2}$$
83 $$1 - 7.27iT - 83T^{2}$$
89 $$1 + 3.37iT - 89T^{2}$$
97 $$1 + 8.55iT - 97T^{2}$$
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}