# Properties

 Degree 2 Conductor $2^{6} \cdot 3^{2} \cdot 7$ Sign $0.338 + 0.941i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−0.0893 + 0.0893i)5-s − 7-s + (−2.42 − 2.42i)11-s + (−3.86 + 3.86i)13-s − 0.794i·17-s + (2.65 + 2.65i)19-s − 3.92i·23-s + 4.98i·25-s + (7.47 + 7.47i)29-s − 5.55i·31-s + (0.0893 − 0.0893i)35-s + (−6.35 − 6.35i)37-s + 6.96·41-s + (1.25 − 1.25i)43-s − 6.48·47-s + ⋯
 L(s)  = 1 + (−0.0399 + 0.0399i)5-s − 0.377·7-s + (−0.731 − 0.731i)11-s + (−1.07 + 1.07i)13-s − 0.192i·17-s + (0.608 + 0.608i)19-s − 0.817i·23-s + 0.996i·25-s + (1.38 + 1.38i)29-s − 0.997i·31-s + (0.0151 − 0.0151i)35-s + (−1.04 − 1.04i)37-s + 1.08·41-s + (0.191 − 0.191i)43-s − 0.945·47-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.338 + 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.338 + 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4032$$    =    $$2^{6} \cdot 3^{2} \cdot 7$$ $$\varepsilon$$ = $0.338 + 0.941i$ motivic weight = $$1$$ character : $\chi_{4032} (3599, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 4032,\ (\ :1/2),\ 0.338 + 0.941i)$ $L(1)$ $\approx$ $1.110771296$ $L(\frac12)$ $\approx$ $1.110771296$ $L(\frac{3}{2})$ 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$$
7 $$1 + T$$
good5 $$1 + (0.0893 - 0.0893i)T - 5iT^{2}$$
11 $$1 + (2.42 + 2.42i)T + 11iT^{2}$$
13 $$1 + (3.86 - 3.86i)T - 13iT^{2}$$
17 $$1 + 0.794iT - 17T^{2}$$
19 $$1 + (-2.65 - 2.65i)T + 19iT^{2}$$
23 $$1 + 3.92iT - 23T^{2}$$
29 $$1 + (-7.47 - 7.47i)T + 29iT^{2}$$
31 $$1 + 5.55iT - 31T^{2}$$
37 $$1 + (6.35 + 6.35i)T + 37iT^{2}$$
41 $$1 - 6.96T + 41T^{2}$$
43 $$1 + (-1.25 + 1.25i)T - 43iT^{2}$$
47 $$1 + 6.48T + 47T^{2}$$
53 $$1 + (0.620 - 0.620i)T - 53iT^{2}$$
59 $$1 + (-3.39 - 3.39i)T + 59iT^{2}$$
61 $$1 + (-7.51 + 7.51i)T - 61iT^{2}$$
67 $$1 + (2.22 + 2.22i)T + 67iT^{2}$$
71 $$1 + 7.95iT - 71T^{2}$$
73 $$1 + 12.9iT - 73T^{2}$$
79 $$1 + 10.1iT - 79T^{2}$$
83 $$1 + (-11.2 + 11.2i)T - 83iT^{2}$$
89 $$1 - 5.52T + 89T^{2}$$
97 $$1 + 4.33T + 97T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}