# Properties

 Label 2-3e4-1.1-c1-0-0 Degree $2$ Conductor $81$ Sign $1$ Analytic cond. $0.646788$ Root an. cond. $0.804231$ Motivic weight $1$ Arithmetic yes Rational no Primitive yes Self-dual yes Analytic rank $0$

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.73·2-s + 0.999·4-s + 1.73·5-s + 2·7-s + 1.73·8-s − 2.99·10-s + 3.46·11-s − 13-s − 3.46·14-s − 5·16-s − 5.19·17-s + 2·19-s + 1.73·20-s − 5.99·22-s − 3.46·23-s − 2.00·25-s + 1.73·26-s + 1.99·28-s − 1.73·29-s + 8·31-s + 5.19·32-s + 9·34-s + 3.46·35-s − 7·37-s − 3.46·38-s + 3.00·40-s + 6.92·41-s + ⋯
 L(s)  = 1 − 1.22·2-s + 0.499·4-s + 0.774·5-s + 0.755·7-s + 0.612·8-s − 0.948·10-s + 1.04·11-s − 0.277·13-s − 0.925·14-s − 1.25·16-s − 1.26·17-s + 0.458·19-s + 0.387·20-s − 1.27·22-s − 0.722·23-s − 0.400·25-s + 0.339·26-s + 0.377·28-s − 0.321·29-s + 1.43·31-s + 0.918·32-s + 1.54·34-s + 0.585·35-s − 1.15·37-s − 0.561·38-s + 0.474·40-s + 1.08·41-s + ⋯

## Functional equation

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

## Invariants

 Degree: $$2$$ Conductor: $$81$$    =    $$3^{4}$$ Sign: $1$ Analytic conductor: $$0.646788$$ Root analytic conductor: $$0.804231$$ Motivic weight: $$1$$ Rational: no Arithmetic: yes Character: $\chi_{81} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 81,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.6236729992$$ $$L(\frac12)$$ $$\approx$$ $$0.6236729992$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$F_p(T)$
bad3 $$1$$
good2 $$1 + 1.73T + 2T^{2}$$
5 $$1 - 1.73T + 5T^{2}$$
7 $$1 - 2T + 7T^{2}$$
11 $$1 - 3.46T + 11T^{2}$$
13 $$1 + T + 13T^{2}$$
17 $$1 + 5.19T + 17T^{2}$$
19 $$1 - 2T + 19T^{2}$$
23 $$1 + 3.46T + 23T^{2}$$
29 $$1 + 1.73T + 29T^{2}$$
31 $$1 - 8T + 31T^{2}$$
37 $$1 + 7T + 37T^{2}$$
41 $$1 - 6.92T + 41T^{2}$$
43 $$1 - 2T + 43T^{2}$$
47 $$1 + 6.92T + 47T^{2}$$
53 $$1 + 53T^{2}$$
59 $$1 + 13.8T + 59T^{2}$$
61 $$1 + 7T + 61T^{2}$$
67 $$1 + 10T + 67T^{2}$$
71 $$1 - 10.3T + 71T^{2}$$
73 $$1 + 7T + 73T^{2}$$
79 $$1 - 2T + 79T^{2}$$
83 $$1 - 13.8T + 83T^{2}$$
89 $$1 - 5.19T + 89T^{2}$$
97 $$1 - 2T + 97T^{2}$$
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$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$

## Imaginary part of the first few zeros on the critical line

−14.24534143580850222689806450921, −13.53005878760745684919879538913, −11.88045000978665525853298817583, −10.81679773430834793125056262951, −9.699851402202700459913494023810, −8.922789416667976924339583277092, −7.76833571382616952776524821959, −6.38667701978104996146814405770, −4.58603454223708864059824147194, −1.78651121087012653708241175100, 1.78651121087012653708241175100, 4.58603454223708864059824147194, 6.38667701978104996146814405770, 7.76833571382616952776524821959, 8.922789416667976924339583277092, 9.699851402202700459913494023810, 10.81679773430834793125056262951, 11.88045000978665525853298817583, 13.53005878760745684919879538913, 14.24534143580850222689806450921