Properties

 Label 2-2e8-4.3-c6-0-44 Degree $2$ Conductor $256$ Sign $-i$ Analytic cond. $58.8938$ Root an. cond. $7.67423$ Motivic weight $6$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 − 46i·3-s − 1.38e3·9-s − 2.33e3i·11-s − 1.72e3·17-s + 2.48e3i·19-s − 1.56e4·25-s + 3.02e4i·27-s − 1.07e5·33-s − 1.34e5·41-s − 7.49e4i·43-s + 1.17e5·49-s + 7.93e4i·51-s + 1.14e5·57-s + 3.04e5i·59-s + 5.96e5i·67-s + ⋯
 L(s)  = 1 − 1.70i·3-s − 1.90·9-s − 1.75i·11-s − 0.351·17-s + 0.361i·19-s − 25-s + 1.53i·27-s − 2.99·33-s − 1.95·41-s − 0.942i·43-s + 49-s + 0.598i·51-s + 0.616·57-s + 1.48i·59-s + 1.98i·67-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(7-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$256$$    =    $$2^{8}$$ Sign: $-i$ Analytic conductor: $$58.8938$$ Root analytic conductor: $$7.67423$$ Motivic weight: $$6$$ Rational: no Arithmetic: yes Character: $\chi_{256} (255, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 256,\ (\ :3),\ -i)$$

Particular Values

 $$L(\frac{7}{2})$$ $$\approx$$ $$0.4851977642$$ $$L(\frac12)$$ $$\approx$$ $$0.4851977642$$ $$L(4)$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad2 $$1$$
good3 $$1 + 46iT - 729T^{2}$$
5 $$1 + 1.56e4T^{2}$$
7 $$1 - 1.17e5T^{2}$$
11 $$1 + 2.33e3iT - 1.77e6T^{2}$$
13 $$1 + 4.82e6T^{2}$$
17 $$1 + 1.72e3T + 2.41e7T^{2}$$
19 $$1 - 2.48e3iT - 4.70e7T^{2}$$
23 $$1 - 1.48e8T^{2}$$
29 $$1 + 5.94e8T^{2}$$
31 $$1 - 8.87e8T^{2}$$
37 $$1 + 2.56e9T^{2}$$
41 $$1 + 1.34e5T + 4.75e9T^{2}$$
43 $$1 + 7.49e4iT - 6.32e9T^{2}$$
47 $$1 - 1.07e10T^{2}$$
53 $$1 + 2.21e10T^{2}$$
59 $$1 - 3.04e5iT - 4.21e10T^{2}$$
61 $$1 + 5.15e10T^{2}$$
67 $$1 - 5.96e5iT - 9.04e10T^{2}$$
71 $$1 - 1.28e11T^{2}$$
73 $$1 - 5.93e5T + 1.51e11T^{2}$$
79 $$1 - 2.43e11T^{2}$$
83 $$1 + 6.78e5iT - 3.26e11T^{2}$$
89 $$1 - 3.57e5T + 4.96e11T^{2}$$
97 $$1 - 1.82e6T + 8.32e11T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$