# Properties

 Degree 2 Conductor $3 \cdot 7$ Sign $-0.907 + 0.420i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (−2.65 − 4.59i)2-s + (1.5 − 2.59i)3-s + (−10.0 + 17.4i)4-s + (−2.78 − 4.81i)5-s − 15.9·6-s + (9.67 − 15.7i)7-s + 64.6·8-s + (−4.5 − 7.79i)9-s + (−14.7 + 25.5i)10-s + (6.95 − 12.0i)11-s + (30.2 + 52.4i)12-s + 38.6·13-s + (−98.2 − 2.58i)14-s − 16.6·15-s + (−90.8 − 157. i)16-s + (−21.7 + 37.6i)17-s + ⋯
 L(s)  = 1 + (−0.938 − 1.62i)2-s + (0.288 − 0.499i)3-s + (−1.26 + 2.18i)4-s + (−0.248 − 0.430i)5-s − 1.08·6-s + (0.522 − 0.852i)7-s + 2.85·8-s + (−0.166 − 0.288i)9-s + (−0.466 + 0.808i)10-s + (0.190 − 0.330i)11-s + (0.728 + 1.26i)12-s + 0.825·13-s + (−1.87 − 0.0492i)14-s − 0.287·15-s + (−1.41 − 2.45i)16-s + (−0.310 + 0.537i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.907 + 0.420i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$21$$    =    $$3 \cdot 7$$ $$\varepsilon$$ = $-0.907 + 0.420i$ motivic weight = $$3$$ character : $\chi_{21} (4, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 21,\ (\ :3/2),\ -0.907 + 0.420i)$ $L(2)$ $\approx$ $0.155592 - 0.705778i$ $L(\frac12)$ $\approx$ $0.155592 - 0.705778i$ $L(\frac{5}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{3,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{3,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad3 $$1 + (-1.5 + 2.59i)T$$
7 $$1 + (-9.67 + 15.7i)T$$
good2 $$1 + (2.65 + 4.59i)T + (-4 + 6.92i)T^{2}$$
5 $$1 + (2.78 + 4.81i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (-6.95 + 12.0i)T + (-665.5 - 1.15e3i)T^{2}$$
13 $$1 - 38.6T + 2.19e3T^{2}$$
17 $$1 + (21.7 - 37.6i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (-54.5 - 94.4i)T + (-3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (-37.4 - 64.8i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 + 72.3T + 2.43e4T^{2}$$
31 $$1 + (32.0 - 55.4i)T + (-1.48e4 - 2.57e4i)T^{2}$$
37 $$1 + (94.3 + 163. i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 + 24.7T + 6.89e4T^{2}$$
43 $$1 + 243.T + 7.95e4T^{2}$$
47 $$1 + (-310. - 537. i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (-143. + 249. i)T + (-7.44e4 - 1.28e5i)T^{2}$$
59 $$1 + (262. - 454. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (-191. - 332. i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (99.0 - 171. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 - 785.T + 3.57e5T^{2}$$
73 $$1 + (-165. + 286. i)T + (-1.94e5 - 3.36e5i)T^{2}$$
79 $$1 + (218. + 379. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 - 241.T + 5.71e5T^{2}$$
89 $$1 + (792. + 1.37e3i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 - 79.2T + 9.12e5T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}