# Properties

 Degree 2 Conductor $3 \cdot 7$ Sign $0.448 - 0.893i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 + (2.27 + 3.94i)2-s + (1.5 − 2.59i)3-s + (−6.38 + 11.0i)4-s + (−8.93 − 15.4i)5-s + 13.6·6-s + (2.26 + 18.3i)7-s − 21.6·8-s + (−4.5 − 7.79i)9-s + (40.7 − 70.5i)10-s + (5.69 − 9.86i)11-s + (19.1 + 33.1i)12-s − 13.0·13-s + (−67.3 + 50.7i)14-s − 53.6·15-s + (1.62 + 2.81i)16-s + (−26.6 + 46.1i)17-s + ⋯
 L(s)  = 1 + (0.805 + 1.39i)2-s + (0.288 − 0.499i)3-s + (−0.797 + 1.38i)4-s + (−0.799 − 1.38i)5-s + 0.930·6-s + (0.122 + 0.992i)7-s − 0.958·8-s + (−0.166 − 0.288i)9-s + (1.28 − 2.23i)10-s + (0.156 − 0.270i)11-s + (0.460 + 0.797i)12-s − 0.279·13-s + (−1.28 + 0.969i)14-s − 0.922·15-s + (0.0254 + 0.0440i)16-s + (−0.379 + 0.658i)17-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.448 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$21$$    =    $$3 \cdot 7$$ $$\varepsilon$$ = $0.448 - 0.893i$ motivic weight = $$3$$ character : $\chi_{21} (4, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 21,\ (\ :3/2),\ 0.448 - 0.893i)$ $L(2)$ $\approx$ $1.27535 + 0.787359i$ $L(\frac12)$ $\approx$ $1.27535 + 0.787359i$ $L(\frac{5}{2})$ 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 + (-2.26 - 18.3i)T$$
good2 $$1 + (-2.27 - 3.94i)T + (-4 + 6.92i)T^{2}$$
5 $$1 + (8.93 + 15.4i)T + (-62.5 + 108. i)T^{2}$$
11 $$1 + (-5.69 + 9.86i)T + (-665.5 - 1.15e3i)T^{2}$$
13 $$1 + 13.0T + 2.19e3T^{2}$$
17 $$1 + (26.6 - 46.1i)T + (-2.45e3 - 4.25e3i)T^{2}$$
19 $$1 + (-21.2 - 36.7i)T + (-3.42e3 + 5.94e3i)T^{2}$$
23 $$1 + (76.0 + 131. i)T + (-6.08e3 + 1.05e4i)T^{2}$$
29 $$1 - 186.T + 2.43e4T^{2}$$
31 $$1 + (-78.9 + 136. i)T + (-1.48e4 - 2.57e4i)T^{2}$$
37 $$1 + (1.87 + 3.24i)T + (-2.53e4 + 4.38e4i)T^{2}$$
41 $$1 + 39.3T + 6.89e4T^{2}$$
43 $$1 - 429.T + 7.95e4T^{2}$$
47 $$1 + (10.5 + 18.3i)T + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + (182. - 316. i)T + (-7.44e4 - 1.28e5i)T^{2}$$
59 $$1 + (-113. + 196. i)T + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (325. + 564. i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (72.7 - 125. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + 368.T + 3.57e5T^{2}$$
73 $$1 + (304. - 527. i)T + (-1.94e5 - 3.36e5i)T^{2}$$
79 $$1 + (455. + 788. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 + 327.T + 5.71e5T^{2}$$
89 $$1 + (-18.8 - 32.5i)T + (-3.52e5 + 6.10e5i)T^{2}$$
97 $$1 - 722.T + 9.12e5T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}