Properties

 Degree 2 Conductor $2 \cdot 3^{3} \cdot 7$ Sign $-0.969 - 0.245i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (0.173 − 0.984i)2-s + (−0.390 − 1.68i)3-s + (−0.939 − 0.342i)4-s + (0.393 − 0.329i)5-s + (−1.72 + 0.0916i)6-s + (−0.939 + 0.342i)7-s + (−0.5 + 0.866i)8-s + (−2.69 + 1.31i)9-s + (−0.256 − 0.444i)10-s + (−2.36 − 1.98i)11-s + (−0.210 + 1.71i)12-s + (−0.932 − 5.29i)13-s + (0.173 + 0.984i)14-s + (−0.710 − 0.534i)15-s + (0.766 + 0.642i)16-s + (−2.52 − 4.37i)17-s + ⋯
 L(s)  = 1 + (0.122 − 0.696i)2-s + (−0.225 − 0.974i)3-s + (−0.469 − 0.171i)4-s + (0.175 − 0.147i)5-s + (−0.706 + 0.0374i)6-s + (−0.355 + 0.129i)7-s + (−0.176 + 0.306i)8-s + (−0.898 + 0.439i)9-s + (−0.0811 − 0.140i)10-s + (−0.714 − 0.599i)11-s + (−0.0606 + 0.496i)12-s + (−0.258 − 1.46i)13-s + (0.0464 + 0.263i)14-s + (−0.183 − 0.138i)15-s + (0.191 + 0.160i)16-s + (−0.612 − 1.06i)17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$378$$    =    $$2 \cdot 3^{3} \cdot 7$$ $$\varepsilon$$ = $-0.969 - 0.245i$ motivic weight = $$1$$ character : $\chi_{378} (295, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 378,\ (\ :1/2),\ -0.969 - 0.245i)$$ $$L(1)$$ $$\approx$$ $$0.103369 + 0.828018i$$ $$L(\frac12)$$ $$\approx$$ $$0.103369 + 0.828018i$$ $$L(\frac{3}{2})$$ 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 + (-0.173 + 0.984i)T$$
3 $$1 + (0.390 + 1.68i)T$$
7 $$1 + (0.939 - 0.342i)T$$
good5 $$1 + (-0.393 + 0.329i)T + (0.868 - 4.92i)T^{2}$$
11 $$1 + (2.36 + 1.98i)T + (1.91 + 10.8i)T^{2}$$
13 $$1 + (0.932 + 5.29i)T + (-12.2 + 4.44i)T^{2}$$
17 $$1 + (2.52 + 4.37i)T + (-8.5 + 14.7i)T^{2}$$
19 $$1 + (2.98 - 5.16i)T + (-9.5 - 16.4i)T^{2}$$
23 $$1 + (-4.07 - 1.48i)T + (17.6 + 14.7i)T^{2}$$
29 $$1 + (-0.663 + 3.76i)T + (-27.2 - 9.91i)T^{2}$$
31 $$1 + (-7.88 - 2.87i)T + (23.7 + 19.9i)T^{2}$$
37 $$1 + (3.24 + 5.62i)T + (-18.5 + 32.0i)T^{2}$$
41 $$1 + (1.73 + 9.83i)T + (-38.5 + 14.0i)T^{2}$$
43 $$1 + (5.49 + 4.61i)T + (7.46 + 42.3i)T^{2}$$
47 $$1 + (-12.7 + 4.63i)T + (36.0 - 30.2i)T^{2}$$
53 $$1 + 2.76T + 53T^{2}$$
59 $$1 + (-4.66 + 3.91i)T + (10.2 - 58.1i)T^{2}$$
61 $$1 + (5.17 - 1.88i)T + (46.7 - 39.2i)T^{2}$$
67 $$1 + (-0.300 - 1.70i)T + (-62.9 + 22.9i)T^{2}$$
71 $$1 + (1.26 + 2.19i)T + (-35.5 + 61.4i)T^{2}$$
73 $$1 + (-4.87 + 8.44i)T + (-36.5 - 63.2i)T^{2}$$
79 $$1 + (1.55 - 8.80i)T + (-74.2 - 27.0i)T^{2}$$
83 $$1 + (-1.50 + 8.54i)T + (-77.9 - 28.3i)T^{2}$$
89 $$1 + (0.964 - 1.67i)T + (-44.5 - 77.0i)T^{2}$$
97 $$1 + (-11.3 - 9.56i)T + (16.8 + 95.5i)T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}