Properties

 Label 2-819-91.54-c1-0-11 Degree $2$ Conductor $819$ Sign $0.452 - 0.891i$ Analytic cond. $6.53974$ Root an. cond. $2.55729$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

Related objects

Dirichlet series

 L(s)  = 1 + (0.926 − 0.926i)2-s + 0.284i·4-s + (−0.409 + 0.109i)5-s + (−2.25 + 1.38i)7-s + (2.11 + 2.11i)8-s + (−0.277 + 0.481i)10-s + (−1.35 + 0.362i)11-s + (−3.54 + 0.648i)13-s + (−0.809 + 3.36i)14-s + 3.35·16-s + 6.94·17-s + (−0.143 + 0.535i)19-s + (−0.0312 − 0.116i)20-s + (−0.918 + 1.59i)22-s + 7.84i·23-s + ⋯
 L(s)  = 1 + (0.654 − 0.654i)2-s + 0.142i·4-s + (−0.183 + 0.0491i)5-s + (−0.852 + 0.522i)7-s + (0.748 + 0.748i)8-s + (−0.0879 + 0.152i)10-s + (−0.408 + 0.109i)11-s + (−0.983 + 0.179i)13-s + (−0.216 + 0.900i)14-s + 0.837·16-s + 1.68·17-s + (−0.0328 + 0.122i)19-s + (−0.00698 − 0.0260i)20-s + (−0.195 + 0.339i)22-s + 1.63i·23-s + ⋯

Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

 Degree: $$2$$ Conductor: $$819$$    =    $$3^{2} \cdot 7 \cdot 13$$ Sign: $0.452 - 0.891i$ Analytic conductor: $$6.53974$$ Root analytic conductor: $$2.55729$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{819} (145, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 819,\ (\ :1/2),\ 0.452 - 0.891i)$$

Particular Values

 $$L(1)$$ $$\approx$$ $$1.36851 + 0.840592i$$ $$L(\frac12)$$ $$\approx$$ $$1.36851 + 0.840592i$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
7 $$1 + (2.25 - 1.38i)T$$
13 $$1 + (3.54 - 0.648i)T$$
good2 $$1 + (-0.926 + 0.926i)T - 2iT^{2}$$
5 $$1 + (0.409 - 0.109i)T + (4.33 - 2.5i)T^{2}$$
11 $$1 + (1.35 - 0.362i)T + (9.52 - 5.5i)T^{2}$$
17 $$1 - 6.94T + 17T^{2}$$
19 $$1 + (0.143 - 0.535i)T + (-16.4 - 9.5i)T^{2}$$
23 $$1 - 7.84iT - 23T^{2}$$
29 $$1 + (-3.01 - 5.22i)T + (-14.5 + 25.1i)T^{2}$$
31 $$1 + (2.17 - 8.09i)T + (-26.8 - 15.5i)T^{2}$$
37 $$1 + (4.24 + 4.24i)T + 37iT^{2}$$
41 $$1 + (-0.434 + 1.62i)T + (-35.5 - 20.5i)T^{2}$$
43 $$1 + (-6.49 - 3.74i)T + (21.5 + 37.2i)T^{2}$$
47 $$1 + (2.62 + 9.79i)T + (-40.7 + 23.5i)T^{2}$$
53 $$1 + (3.77 + 6.54i)T + (-26.5 + 45.8i)T^{2}$$
59 $$1 + (-4.83 + 4.83i)T - 59iT^{2}$$
61 $$1 + (-2.38 + 1.37i)T + (30.5 - 52.8i)T^{2}$$
67 $$1 + (-3.70 - 13.8i)T + (-58.0 + 33.5i)T^{2}$$
71 $$1 + (1.00 + 3.74i)T + (-61.4 + 35.5i)T^{2}$$
73 $$1 + (11.0 + 2.96i)T + (63.2 + 36.5i)T^{2}$$
79 $$1 + (-4.32 + 7.49i)T + (-39.5 - 68.4i)T^{2}$$
83 $$1 + (-2.26 - 2.26i)T + 83iT^{2}$$
89 $$1 + (-8.19 + 8.19i)T - 89iT^{2}$$
97 $$1 + (-15.5 + 4.17i)T + (84.0 - 48.5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$