# Properties

 Label 2-3e4-9.4-c3-0-7 Degree $2$ Conductor $81$ Sign $0.173 + 0.984i$ Analytic cond. $4.77915$ Root an. cond. $2.18612$ Motivic weight $3$ Arithmetic yes Rational no Primitive yes Self-dual no Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 + (4 − 6.92i)4-s + (−10 − 17.3i)7-s + (35 − 60.6i)13-s + (−31.9 − 55.4i)16-s + 56·19-s + (62.5 + 108. i)25-s − 160·28-s + (−154 + 266. i)31-s + 110·37-s + (260 + 450. i)43-s + (−28.5 + 49.3i)49-s + (−279. − 484. i)52-s + (−91 − 157. i)61-s − 511.·64-s + (440 − 762. i)67-s + ⋯
 L(s)  = 1 + (0.5 − 0.866i)4-s + (−0.539 − 0.935i)7-s + (0.746 − 1.29i)13-s + (−0.499 − 0.866i)16-s + 0.676·19-s + (0.5 + 0.866i)25-s − 1.07·28-s + (−0.892 + 1.54i)31-s + 0.488·37-s + (0.922 + 1.59i)43-s + (−0.0830 + 0.143i)49-s + (−0.746 − 1.29i)52-s + (−0.191 − 0.330i)61-s − 0.999·64-s + (0.802 − 1.38i)67-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(4-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$81$$    =    $$3^{4}$$ Sign: $0.173 + 0.984i$ Analytic conductor: $$4.77915$$ Root analytic conductor: $$2.18612$$ Motivic weight: $$3$$ Rational: no Arithmetic: yes Character: $\chi_{81} (28, \cdot )$ Primitive: yes Self-dual: no Analytic rank: $$0$$ Selberg data: $$(2,\ 81,\ (\ :3/2),\ 0.173 + 0.984i)$$

## Particular Values

 $$L(2)$$ $$\approx$$ $$1.16433 - 0.976993i$$ $$L(\frac12)$$ $$\approx$$ $$1.16433 - 0.976993i$$ $$L(\frac{5}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
good2 $$1 + (-4 + 6.92i)T^{2}$$
5 $$1 + (-62.5 - 108. i)T^{2}$$
7 $$1 + (10 + 17.3i)T + (-171.5 + 297. i)T^{2}$$
11 $$1 + (-665.5 + 1.15e3i)T^{2}$$
13 $$1 + (-35 + 60.6i)T + (-1.09e3 - 1.90e3i)T^{2}$$
17 $$1 + 4.91e3T^{2}$$
19 $$1 - 56T + 6.85e3T^{2}$$
23 $$1 + (-6.08e3 - 1.05e4i)T^{2}$$
29 $$1 + (-1.21e4 + 2.11e4i)T^{2}$$
31 $$1 + (154 - 266. i)T + (-1.48e4 - 2.57e4i)T^{2}$$
37 $$1 - 110T + 5.06e4T^{2}$$
41 $$1 + (-3.44e4 - 5.96e4i)T^{2}$$
43 $$1 + (-260 - 450. i)T + (-3.97e4 + 6.88e4i)T^{2}$$
47 $$1 + (-5.19e4 + 8.99e4i)T^{2}$$
53 $$1 + 1.48e5T^{2}$$
59 $$1 + (-1.02e5 - 1.77e5i)T^{2}$$
61 $$1 + (91 + 157. i)T + (-1.13e5 + 1.96e5i)T^{2}$$
67 $$1 + (-440 + 762. i)T + (-1.50e5 - 2.60e5i)T^{2}$$
71 $$1 + 3.57e5T^{2}$$
73 $$1 - 1.19e3T + 3.89e5T^{2}$$
79 $$1 + (442 + 765. i)T + (-2.46e5 + 4.26e5i)T^{2}$$
83 $$1 + (-2.85e5 + 4.95e5i)T^{2}$$
89 $$1 + 7.04e5T^{2}$$
97 $$1 + (-665 - 1.15e3i)T + (-4.56e5 + 7.90e5i)T^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$