Properties

 Degree 2 Conductor $2^{4}$ Sign $0.690 - 0.723i$ Motivic weight 3 Primitive yes Self-dual no Analytic rank 0

Related objects

Dirichlet series

 L(s)  = 1 + (0.836 + 2.70i)2-s + (1.98 − 1.98i)3-s + (−6.59 + 4.52i)4-s + (−0.596 − 0.596i)5-s + (7.01 + 3.69i)6-s − 29.0i·7-s + (−17.7 − 14.0i)8-s + 19.1i·9-s + (1.11 − 2.11i)10-s + (12.1 + 12.1i)11-s + (−4.11 + 22.0i)12-s + (−48.5 + 48.5i)13-s + (78.5 − 24.3i)14-s − 2.36·15-s + (23.0 − 59.6i)16-s + 86.7·17-s + ⋯
 L(s)  = 1 + (0.295 + 0.955i)2-s + (0.381 − 0.381i)3-s + (−0.824 + 0.565i)4-s + (−0.0533 − 0.0533i)5-s + (0.477 + 0.251i)6-s − 1.57i·7-s + (−0.784 − 0.620i)8-s + 0.708i·9-s + (0.0351 − 0.0667i)10-s + (0.332 + 0.332i)11-s + (−0.0991 + 0.530i)12-s + (−1.03 + 1.03i)13-s + (1.50 − 0.464i)14-s − 0.0407·15-s + (0.360 − 0.932i)16-s + 1.23·17-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.690 - 0.723i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 $$d$$ = $$2$$ $$N$$ = $$16$$    =    $$2^{4}$$ $$\varepsilon$$ = $0.690 - 0.723i$ motivic weight = $$3$$ character : $\chi_{16} (5, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 16,\ (\ :3/2),\ 0.690 - 0.723i)$$ $$L(2)$$ $$\approx$$ $$1.05836 + 0.452995i$$ $$L(\frac12)$$ $$\approx$$ $$1.05836 + 0.452995i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \neq 2$,$$F_p(T)$$ is a polynomial of degree 2. If $p = 2$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 + (-0.836 - 2.70i)T$$
good3 $$1 + (-1.98 + 1.98i)T - 27iT^{2}$$
5 $$1 + (0.596 + 0.596i)T + 125iT^{2}$$
7 $$1 + 29.0iT - 343T^{2}$$
11 $$1 + (-12.1 - 12.1i)T + 1.33e3iT^{2}$$
13 $$1 + (48.5 - 48.5i)T - 2.19e3iT^{2}$$
17 $$1 - 86.7T + 4.91e3T^{2}$$
19 $$1 + (54.8 - 54.8i)T - 6.85e3iT^{2}$$
23 $$1 + 70.2iT - 1.21e4T^{2}$$
29 $$1 + (-63.4 + 63.4i)T - 2.43e4iT^{2}$$
31 $$1 + 8.86T + 2.97e4T^{2}$$
37 $$1 + (21.7 + 21.7i)T + 5.06e4iT^{2}$$
41 $$1 - 153. iT - 6.89e4T^{2}$$
43 $$1 + (120. + 120. i)T + 7.95e4iT^{2}$$
47 $$1 + 99.9T + 1.03e5T^{2}$$
53 $$1 + (-389. - 389. i)T + 1.48e5iT^{2}$$
59 $$1 + (324. + 324. i)T + 2.05e5iT^{2}$$
61 $$1 + (0.339 - 0.339i)T - 2.26e5iT^{2}$$
67 $$1 + (-565. + 565. i)T - 3.00e5iT^{2}$$
71 $$1 - 419. iT - 3.57e5T^{2}$$
73 $$1 + 374. iT - 3.89e5T^{2}$$
79 $$1 + 705.T + 4.93e5T^{2}$$
83 $$1 + (947. - 947. i)T - 5.71e5iT^{2}$$
89 $$1 - 4.72iT - 7.04e5T^{2}$$
97 $$1 - 379.T + 9.12e5T^{2}$$
\begin{aligned}L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\end{aligned}