# Properties

 Label 2-3e3-1.1-c1-0-0 Degree $2$ Conductor $27$ Sign $1$ Analytic cond. $0.215596$ Root an. cond. $0.464323$ Motivic weight $1$ Arithmetic yes Rational yes Primitive yes Self-dual yes Analytic rank $0$

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## Dirichlet series

 L(s)  = 1 − 2·4-s − 7-s + 5·13-s + 4·16-s − 7·19-s − 5·25-s + 2·28-s − 4·31-s + 11·37-s + 8·43-s − 6·49-s − 10·52-s − 61-s − 8·64-s + 5·67-s − 7·73-s + 14·76-s + 17·79-s − 5·91-s − 19·97-s + 10·100-s − 13·103-s + 2·109-s − 4·112-s + ⋯
 L(s)  = 1 − 4-s − 0.377·7-s + 1.38·13-s + 16-s − 1.60·19-s − 25-s + 0.377·28-s − 0.718·31-s + 1.80·37-s + 1.21·43-s − 6/7·49-s − 1.38·52-s − 0.128·61-s − 64-s + 0.610·67-s − 0.819·73-s + 1.60·76-s + 1.91·79-s − 0.524·91-s − 1.92·97-s + 100-s − 1.28·103-s + 0.191·109-s − 0.377·112-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$2$$ Conductor: $$27$$    =    $$3^{3}$$ Sign: $1$ Analytic conductor: $$0.215596$$ Root analytic conductor: $$0.464323$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: $\chi_{27} (1, \cdot )$ Primitive: yes Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(2,\ 27,\ (\ :1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$0.5888795834$$ $$L(\frac12)$$ $$\approx$$ $$0.5888795834$$ $$L(\frac{3}{2})$$ 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 + p T^{2}$$
5 $$1 + p T^{2}$$
7 $$1 + T + p T^{2}$$
11 $$1 + p T^{2}$$
13 $$1 - 5 T + p T^{2}$$
17 $$1 + p T^{2}$$
19 $$1 + 7 T + p T^{2}$$
23 $$1 + p T^{2}$$
29 $$1 + p T^{2}$$
31 $$1 + 4 T + p T^{2}$$
37 $$1 - 11 T + p T^{2}$$
41 $$1 + p T^{2}$$
43 $$1 - 8 T + p T^{2}$$
47 $$1 + p T^{2}$$
53 $$1 + p T^{2}$$
59 $$1 + p T^{2}$$
61 $$1 + T + p T^{2}$$
67 $$1 - 5 T + p T^{2}$$
71 $$1 + p T^{2}$$
73 $$1 + 7 T + p T^{2}$$
79 $$1 - 17 T + p T^{2}$$
83 $$1 + p T^{2}$$
89 $$1 + p T^{2}$$
97 $$1 + 19 T + p T^{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

−17.54958210256512249523818407718, −16.30001713525188163931408346208, −14.89211076740924954992271086519, −13.58211369833951574831601484907, −12.71563949069013251763955822657, −10.90872829298908742564232271224, −9.429199208210393651255146412199, −8.217650367462526737991465554229, −6.04893540000987436402649531985, −4.04304401379743272242521882180, 4.04304401379743272242521882180, 6.04893540000987436402649531985, 8.217650367462526737991465554229, 9.429199208210393651255146412199, 10.90872829298908742564232271224, 12.71563949069013251763955822657, 13.58211369833951574831601484907, 14.89211076740924954992271086519, 16.30001713525188163931408346208, 17.54958210256512249523818407718